Logarithmic Function Manipulation

Lesson ~10 min read 8 MCQs

In simple terms: In simple terms, logarithmic properties are rules that let you rewrite complicated log expressions into simpler ones, much like rearranging furniture in a room to see it in a new way.

Why this matters

Have you ever used photo editing software, like on your phone or a school computer? You move a slider for "brightness" and the whole image gets lighter. You move another for "contrast" and the darks get darker while the lights get lighter.

What's fascinating is that sometimes, two different sliders can produce the exact same result. Maybe sliding "exposure" up a little bit looks identical to sliding "brightness" up a lot.

Logarithms work in a surprisingly similar way. The rules we're about to explore are like discovering that a "horizontal squish" on a log graph is secretly the same as a "vertical shift." By learning to manipulate these functions, you're not just shuffling symbols around—you're gaining a deeper understanding of how functions transform, which is a huge idea in all of math. Let's dive in.

Concept overview

flowchart TD
    A[Logarithmic Expression] --> B{Goal?};
    B --> C[Expand];
    B --> D[Condense];

    C --> C1{Has a quotient?};
    C1 -- Yes --> C2[Use Quotient Rule: log(A/B) = log(A) - log(B)];
    C1 -- No --> C3;
    C2 --> C3{Has a product?};
    C3 -- Yes --> C4[Use Product Rule: log(AB) = log(A) + log(B)];
    C3 -- No --> C5;
    C4 --> C5{Has a power?};
    C5 -- Yes --> C6[Use Power Rule: log(A^n) = n*log(A)];
    C5 -- No --> C7[Done];
    C6 --> C7;

    D --> D1{Has coefficients?};
    D1 -- Yes --> D2[Use Power Rule in reverse];
    D1 -- No --> D3;
    D2 --> D3{Has sums/differences?};
    D3 -- Yes --> D4[Use Product/Quotient Rules in reverse];
    D4 --> D5[Done];
    D3 -- No --> D5;

Core explanation

Alright, let's get to it. Before we can manipulate logarithms, let's do a quick refresh. A logarithm is the inverse of an exponential function. It answers the question: "What exponent do I need to raise the base to, to get this number?"

For example, log₂ 8 = 3 because 2³ = 8.

The Natural Logarithm: A Special Case

You'll see ln x all the time in this course and in future math and science classes. Don't let it intimidate you.

The function f(x) = ln x is simply a logarithm with a special base called e. That is, ln x is shorthand for logₑ x.

What's e? It's an irrational number, approximately 2.718, kind of like how π is approximately 3.14. It's called the "natural" base because it shows up everywhere in nature, finance, and physics. For our purposes today, just treat ln x like any other logarithm, but with a base of e.

Property 1: The Product Property

This rule helps you handle the logarithm of a multiplication.

Product Property: log_b (xy) = log_b x + log_b y

In words: The log of a product is the sum of the logs.

The Graph Connection (EK 2.12.A.1) This is where it gets really cool. Let's look at the function g(x) = ln(2x). This is a horizontal compression of the parent function f(x) = ln(x) by a factor of 1/2. The graph gets squished toward the y-axis.

But wait! According to the product property: ln(2x) = ln(2) + ln(x)

ln(2) is just a number (it's about 0.693). So, we've rewritten the function as: g(x) = ln(x) + 0.693

This is a vertical translation of the parent function f(x) = ln(x) up by ln(2) units.

This is a huge idea: a horizontal compression of a log graph is identical to a vertical shift. The product property is the algebraic proof of this graphical fact.

Property 2: The Power Property

This rule tells you what to do when the input of your logarithm is raised to a power.

Power Property: log_b (xⁿ) = n ⋅ log_b x

In words: An exponent on the input can be brought down as a multiplier in front of the logarithm.

The Graph Connection (EK 2.12.A.2) Let's consider the function p(x) = ln(x²). Here, we're squaring the input before taking the log.

Using the power property, we can rewrite this: ln(x²) = 2 ⋅ ln(x)

So, the function p(x) = ln(x²) is identical to the function q(x) = 2 ln(x).

What does q(x) = 2 ln(x) represent? It's a vertical dilation (a stretch) of the parent function f(x) = ln(x) by a factor of 2. Every y-value gets doubled.

So, raising the input of a log function to a power is the same as vertically stretching the graph of the original log function.

Property 3: The Change of Base Property

Your calculator has a log button (which is base 10) and an ln button (base e). But what if you need to calculate log₄(25)? You need a way to change the base to something your calculator understands.

Change of Base Property: log_b x = (logₐ x) / (logₐ b)

You can change to any new base a (as long as it's positive and not 1), but we almost always choose base 10 or base e.

So, to find log₄(25): log₄(25) = ln(25) / ln(4) ≈ 3.219 / 1.386 ≈ 2.32 OR log₄(25) = log(25) / log(4) ≈ 1.398 / 0.602 ≈ 2.32 You get the same answer either way!

The Graph Connection (EK 2.12.A.3) Let's look at the formula again: log_b x = (1 / logₐ b) ⋅ logₐ x.

The term (1 / logₐ b) is just a constant number. Let's call it c. So, log_b x = c ⋅ logₐ x.

This stunningly reveals that any logarithmic function is just a vertical dilation of any other logarithmic function. The graph of y = log₄(x) has the same fundamental shape as y = ln(x); it's just been vertically stretched or compressed by a specific factor. They are all part of the same family.

A horizontal compression of ln(x) is equivalent to a vertical shift.

Worked examples

Let's put these properties into practice. This is how you'll use them on homework and the AP exam.

Example 1

Expanding a Logarithmic Expression

Problem: Expand the expression log₃( (9x²y) / z³ ) into a sum and/or difference of logarithms.

Solution: Our goal is to break this single, complicated logarithm down into simpler pieces. We'll handle one rule at a time, working from the outside in.

1
Address the division (Quotient Rule)
The main structure is a fraction. The quotient rule says log(A/B) = log(A) - log(B). log₃(9x²y) - log₃(z³) Why? We're separating the numerator and the denominator. This is our first big step to unpacking the expression.
2
Address the multiplication (Product Rule)
Now look at the first term, log₃(9x²y). The input 9x²y is a product of three things. The product rule lets us split this into a sum. (log₃(9) + log₃(x²) + log₃(y)) - log₃(z³) Why? We've broken down the product inside the first log. I've kept the parentheses to be extra clear about what came from that first term.
3
Address the exponents (Power Rule)
We have two terms with exponents: log₃(x²) and log₃(z³). The power rule lets us bring those exponents down in front as multipliers. log₃(9) + 2⋅log₃(x) + log₃(y) - 3⋅log₃(z) Why? This is the final "unpacking" step. The exponents are now coefficients.
4
Simplify any numerical logs
Can we simplify log₃(9)? Yes! The question is "3 to what power is 9?" The answer is 2. 2 + 2⋅log₃(x) + log₃(y) - 3⋅log₃(z)

This is our final, fully expanded answer.

Example 2

Condensing a Logarithmic Expression

Problem: Write the expression 2 ln(x) - ln(x+5) + (1/2)ln(y) as a single logarithm.

Solution: Now we're doing the reverse. We're "repacking" the expression.

Address the coefficients (Power Rule in reverse): The power rule n log(x) = log(xⁿ) works both ways. Any number multiplying a log can become an exponent on the input. ln(x²) - ln(x+5) + ln(y¹/²) Why? To combine logs using the product/quotient rules, they can't have any coefficients. This step clears the way. Remember that an exponent of 1/2 means a square root. ln(x²) - ln(x+5) + ln(√y)
Address the addition and subtraction (Product/Quotient Rules): Work from left to right. Terms that are added will be multiplied in the numerator. Terms that are subtracted will be divided. The ln(x²) is positive (it's a product in the numerator). The ln(x+5) is subtracted (it's a division in the denominator). The ln(√y) is added (it's a product in the numerator).

So, we combine them all into one ln: ln( (x² ⋅ √y) / (x+5) )

This is the final, condensed answer. This is where most students slip up: they might incorrectly place the √y in the denominator. Just remember: plus signs mean multiplication, minus signs mean division.

Visualizing the transformation from ln(x) to ln(x/e).

Try it yourself

Time to try it on your own. Don't worry about getting it perfect the first time; focus on applying the process.

1
Condense this expression
3 log(x) + 4 log(y) - log(z) Hint: Start by moving the coefficients 3 and 4 back up into the exponent position. Then, combine the terms using the product and quotient rules.
2
Rewrite and describe the transformation
The function g(x) = ln(x/e) is a transformation of the parent function f(x) = ln(x). Use the log properties to rewrite g(x) and describe the transformation in words. Hint: Use the quotient rule first. What is ln(e)? Remember, ln is log base e. So logₑ(e) is asking, "e to what power equals e?"

TL;DR

In simple terms, logarithmic properties are rules that let you rewrite complicated log expressions into simpler ones, much like rearranging furniture in a room to see it in a new way.

Key terms

AP Precalculus Logarithmic functions Log properties Product rule Power rule Quotient rule Change of base Natural logarithm Ln(x) Graph transformations ===END===

You can now…

2.12.A: Rewrite logarithmic expressions in equivalent forms.

Essential knowledge (exam-tested)

2.12.A.1: The product property for logarithms states that log_b (xy) = log_b x + log_b y. Graphically, this property implies that every horizontal dilation of a logarithmic function, f (x) = log_b (kx), is equivalent to a vertical translation, f(x) = log_b (kx) = log_b k + log_b x = a + log_b x, where a = log_b k.
2.12.A.2: The power property for logarithms states that log_b x^n = n log_b x. Graphically, this property implies that raising the input of a logarithmic function to a power, f (x) = log_b x^k, results in a vertical dilation, f (x) = log_b x^k = k log_b x.
2.12.A.3: The change of base property for logarithms states that log_b x = (log_a x) / (log_a b), where a > 0 and a ≠ 1. This implies that all logarithmic functions are vertical dilations of each other.
2.12.A.4: The function f (x) = ln x is a logarithmic function with the natural base e; that is, ln x = log_e x.

Concept map

flowchart TD
    A[Logarithmic Expression] --> B{Goal?};
    B --> C[Expand];
    B --> D[Condense];

    C --> C1{Has a quotient?};
    C1 -- Yes --> C2[Use Quotient Rule: log(A/B) = log(A) - log(B)];
    C1 -- No --> C3;
    C2 --> C3{Has a product?};
    C3 -- Yes --> C4[Use Product Rule: log(AB) = log(A) + log(B)];
    C3 -- No --> C5;
    C4 --> C5{Has a power?};
    C5 -- Yes --> C6[Use Power Rule: log(A^n) = n*log(A)];
    C5 -- No --> C7[Done];
    C6 --> C7;

    D --> D1{Has coefficients?};
    D1 -- Yes --> D2[Use Power Rule in reverse];
    D1 -- No --> D3;
    D2 --> D3{Has sums/differences?};
    D3 -- Yes --> D4[Use Product/Quotient Rules in reverse];
    D4 --> D5[Done];
    D3 -- No --> D5;

Read what Saavi narrates

Hey there. It's Saavi.

Have you ever messed around with the editing tools on a photo? You know, you can slide the "brightness" up, or maybe you slide the "exposure" up, and sometimes you can get the exact same look from two different settings. It feels like you've found a little secret in the software.

That's exactly what we're doing today with logarithms. We're going to learn the secret rules that let us manipulate log expressions. It's not just about shuffling symbols—it's about seeing that a horizontal squish on a graph can be the exact same thing as a vertical shift. It's a really powerful idea.

We'll focus on three main properties: the Product, Power, and Change of Base rules. Think of these as your toolkit for repacking and unpacking logarithms to make them easier to work with.

Let's try one. Imagine you're asked to expand the expression log base three of the fraction, nine times x-squared times y, all over z-cubed. It sounds like a lot, but we just take it one step at a time.

First, we see the big fraction. The quotient rule tells us we can split that into two logs being subtracted. So we get log base three of the numerator, minus log base three of the denominator. That's log base three of nine x-squared y, minus log base three of z-cubed.

Now, we look at that first term. It's a product. The product rule lets us split it into a sum of logs. So we have log base three of nine, plus log base three of x-squared, plus log base three of y. And we still have that minus log base three of z-cubed at the end.

Almost there. Now we use the power rule. Any exponents can come down in front as multipliers. So the log of x-squared becomes two times the log of x. And the log of z-cubed becomes three times the log of z.

Our final step is to see if we can simplify. What's log base three of nine? Well, three to what power is nine? It's two. So we replace that first term with the number 2.

Our final, expanded expression is: two, plus two times log base three of x, plus log base three of y, minus three times log base three of z. We took something messy and made it neat.

Now, one of the biggest mistakes I see every year is students trying to "distribute" a logarithm. They see log of (x plus y) and they write log of x plus log of y. Please, please remember: that's not a rule! The product rule is for the log of a product, x times y, not a sum. Log of (x plus y) can't be simplified. It's a trap!

You're building a really important skill here. It takes practice, but you are more than capable of mastering this. Keep up the great work.

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