---
title: "Change of Base Formula — AP Precalc Definition & Guide"
description: "The change of base formula, log_b(x) = log_c(x)/log_c(b), lets you rewrite any logarithm in a new base. It's how you solve equations like 3^x = 50 in Topic 2.13."
canonical: "https://fiveable.me/ap-pre-calc/key-terms/change-of-base-formula"
type: "key-term"
subject: "AP Pre-Calculus"
unit: "Unit 2"
---

# Change of Base Formula — AP Precalc Definition & Guide

## Definition

The change of base formula is the logarithmic identity log_b(x) = log_c(x) / log_c(b), which rewrites a logarithm in any base b as a ratio of logarithms in a base c you choose. In AP Precalculus Topic 2.13, it lets you evaluate logs your calculator can't compute directly and rewrite exponential expressions in new bases.

## What It Is

The change of base formula says log_b(x) = log_c(x) / log_c(b). In plain terms, if you need a [logarithm](/ap-pre-calc/key-terms/logarithm "fv-autolink") in an inconvenient [base](/ap-pre-calc/unit-2/inverses-exponential-functions/study-guide/7mdx6zi19alJ4hK3 "fv-autolink") b, you can swap it for a fraction of two logarithms in any base c you like, usually base 10 or base e, since those are the ones your calculator has buttons for. So log_3(50) becomes ln(50)/ln(3), and suddenly an unsolvable-looking number is just a quick calculation.

The formula has a twin that shows up in the CED for Topic 2.13. The essential knowledge for [[AP Pre Calc](/ap-pre-calc "fv-autolink") 2.13.A] states that b^x = c^((log_c b)(x)). That identity is change of base for exponential functions. It says any exponential function can be rewritten with a different base, as long as you scale the exponent by log_c(b). Both versions are doing the same job, translating between bases so you can pick whichever base makes the problem easiest.

## Why It Matters

This lives in **[Unit 2](/ap-pre-calc/unit-2 "fv-autolink"): Exponential and Logarithmic Functions**, specifically **Topic 2.13: Exponential and Logarithmic Equations and Inequalities**, supporting [AP Pre Calc 2.13.A] (solve exponential and logarithmic equations and inequalities). The CED says [properties of logarithms](/ap-pre-calc/key-terms/properties-of-logarithms "fv-autolink") are tools for solving these equations, and change of base is the property that bridges the gap between the base a problem hands you and the base you can actually work with. An equation like 3^x = 50 can't be solved by matching bases, so you take a log of both sides and end up with x = log(50)/log(3), which is the change of base formula in action. The exponential version, b^x = c^((log_c b)(x)), also matters because rewriting an exponential in base e or base 2 can reveal helpful information about a model, which is exactly the kind of rewriting the CED calls out.

## Connections

### [Properties of logarithms (Unit 2)](/ap-pre-calc/key-terms/properties-of-logarithms)

Change of base is one of the log properties, and it teams up with the product, [quotient](/ap-pre-calc/unit-1/equivalent-representations-polynomial-rational-expressions/study-guide/NRzwc7vjmULoqIyP "fv-autolink"), and power rules constantly. A typical solve uses the power rule to pull an exponent down, then change of base to get a number out of the result.

### Exponential equations (Unit 2)

When an [exponential equation](/ap-pre-calc/key-terms/exponential-equation "fv-autolink") has mismatched bases, like 3^x = 50, change of base is the exit strategy. You take a log of both sides and the answer naturally lands in the form log(50)/log(3).

### [Properties of exponents (Unit 2)](/ap-pre-calc/key-terms/properties-of-exponents)

The identity b^x = c^((log_c b)(x)) only works because of [exponent rules](/ap-pre-calc/key-terms/exponent-rules "fv-autolink"), since (c^(log_c b))^x = c^((log_c b)x). Change of base for exponentials is really an exponent property and a log property shaking hands.

### [Extraneous solutions (Unit 2)](/ap-pre-calc/key-terms/extraneous-solutions)

After you change bases and solve, the CED still expects you to check your answer against domain restrictions. Logarithms only accept positive inputs, so a solution that makes any log's argument zero or negative gets thrown out.

## On the AP Exam

Multiple-choice questions hit this two ways. First, the classic solve. A question gives you something like 3^x = 50, points out that you can't match the bases, and asks which method works. The answer is to take a logarithm of both sides, which produces x = log(50)/log(3). Second, the identity version. Questions test whether you recognize that c^(log_c b) = b, asking things like "if e^(kx) = (2^(log_2 e))^(kx), what is log_2 e?" These look intimidating but collapse fast once you see that 2^(log_2 e) is just e. No released FRQ has used the phrase "change of base formula" verbatim, but Part A (calculator-allowed) FRQs routinely involve solving exponential equations where your calculator answer comes from exactly this formula. Know it both directions, log form and exponential form.

## change of base formula vs the quotient rule for logarithms

Change of base says log_b(x) = log_c(x)/log_c(b), a ratio of two separate logs. The quotient rule says log_c(x/b) = log_c(x) - log_c(b), one log of a fraction turning into a difference. Mixing them up is a classic trap. log(50)/log(3) is NOT log(50/3) and it's NOT log(50) - log(3). Division of logs and logs of division are completely different things.

## Key Takeaways

- The change of base formula is log_b(x) = log_c(x)/log_c(b), and the new base c can be anything, though base 10 or base e is the practical choice for calculators.
- Its exponential twin, b^x = c^((log_c b)(x)), appears in the CED for Topic 2.13 and lets you rewrite any exponential function in a new base.
- The shortcut identity c^(log_c b) = b is the engine behind change of base, and AP multiple-choice questions test whether you spot it inside scary-looking expressions.
- An equation like 3^x = 50 can't be solved by matching bases, so you take a log of both sides and the solution comes out as log(50)/log(3).
- Dividing two logs is not the same as the log of a quotient. log(50)/log(3) does not equal log(50/3) or log(50) - log(3).
- After changing bases and solving, check for extraneous solutions, since log arguments must stay positive.

## FAQs

### What is the change of base formula in AP Precalculus?

It's the identity log_b(x) = log_c(x)/log_c(b), which rewrites a logarithm in base b as a ratio of logarithms in a new base c. It supports learning objective 2.13.A in Unit 2, solving exponential and logarithmic equations.

### Is log(50)/log(3) the same as log(50/3)?

No. log(50)/log(3) is the change of base formula and equals log_3(50), roughly 3.56. log(50/3) uses the quotient rule and is a totally different number, about 1.22. Division of logs and logs of division are not interchangeable.

### Do I need to memorize the change of base formula for the AP Precalc exam?

Yes. The exam doesn't give you a formula sheet of log identities, and the CED for Topic 2.13 explicitly includes the related identity b^x = c^((log_c b)(x)). You should be able to use both the log version and the exponential version.

### How is change of base different from the quotient rule for logs?

Change of base turns one log into a fraction of two logs in a new base, log_b(x) = log_c(x)/log_c(b). The quotient rule turns the log of a fraction into a difference, log_c(x/b) = log_c(x) - log_c(b). One outputs division, the other outputs subtraction, and they answer different questions.

### How do you solve 3^x = 50 with the change of base formula?

You can't write both sides with the same base, so take a logarithm of both sides. That gives x·log(3) = log(50), so x = log(50)/log(3) ≈ 3.56. The result is literally log_3(50) rewritten by change of base.

## Related Study Guides

- [2.13 Exponential and Logarithmic Equations and Inequalities](/ap-pre-calc/unit-2/exponential-logarithmic-equations-inequalities/study-guide/Mor8iJ1w4VWPX8Wi)

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