Practice Change of Base Formula in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

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Quick Recap

A formula for converting a logarithm from one base to another: log⁑bx=ln⁑xln⁑b=log⁑xlog⁑b\log_b x = \frac{\ln x}{\ln b} = \frac{\log x}{\log b}.

Your calculator only has ln⁑\ln and log⁑10\log_{10} buttons. The change-of-base formula lets you compute ANY logarithm using whichever base you have available. It works because all logarithms are proportional to each otherβ€”changing base just changes the scale factor.

Showing a random 20 of 50 problems.

Example 1

easy
Evaluate log⁑264\log_2 64 without a calculator.

Example 2

challenge
If log⁑2x=a\log_2 x = a, express log⁑8x\log_8 x and log⁑4x\log_4 x in terms of aa.

Example 3

hard
If log⁑25=a\log_2 5 = a, express log⁑532\log_5 32 in terms of aa.

Example 4

medium
Solve log⁑4x=log⁑28\log_4 x = \log_2 8 for xx.

Example 5

medium
Evaluate log⁑464\log_4 64 exactly.

Example 6

easy
Evaluate log⁑51\log_5 1.

Example 7

hard
Prove that log⁑a(b)β‹…log⁑b(c)=log⁑a(c)\log_a(b) \cdot \log_b(c) = \log_a(c) using the change of base formula.

Example 8

medium
Simplify log⁑168\log_{16} 8 to an exact fraction.

Example 9

medium
Show that log⁑ba=1log⁑ab\log_b a = \dfrac{1}{\log_a b} using change-of-base.

Example 10

easy
Rewrite log⁑125\log_{12} 5 as a ratio of common (base-10) logarithms.

Example 11

easy
Evaluate log⁑bb\log_b b for any base b>0b>0, bβ‰ 1b\ne 1.

Example 12

medium
Simplify log⁑9(27)\log_9(27) to an exact fraction.

Example 13

easy
Use change-of-base to write log⁑4x\log_4 x in terms of base 2 logs.

Example 14

hard
Solve log⁑2x+log⁑4x=6\log_2 x + \log_4 x = 6 for x>0x > 0.

Example 15

medium
Evaluate log⁑12525\log_{125} 25 exactly.

Example 16

hard
If log⁑32=p\log_3 2 = p and log⁑35=q\log_3 5 = q, find log⁑56\log_5 6 in terms of pp and qq.

Example 17

easy
Evaluate log⁑232\log_2 32 by recognizing a power.

Example 18

challenge
Solve log⁑2x+log⁑4x=3\log_2 x + \log_4 x = 3 for xx.

Example 19

medium
Evaluate log⁑23β‹…log⁑38\log_2 3 \cdot \log_3 8 exactly.

Example 20

easy
Rewrite log⁑750\log_7 50 as a ratio of natural logs.
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