\( \newcommand{\abs}[1]{\left| \, {#1} \, \right| } \) \( \newcommand{\cm}{\mathrm{cm} } \) \( \newcommand{\sec}{ \, \mathrm{sec} \, } \) \( \newcommand{\units}[1]{\,\text{#1}} \) \( \newcommand{\vhat}[1]{\,\hat{#1}} \) \( \newcommand{\vhati}{\,\hat{i}} \) \( \newcommand{\vhatj}{\,\hat{j}} \) \( \newcommand{\vhatk}{\,\hat{k}} \) \( \newcommand{\vect}[1]{\boldsymbol{\vec{#1}}} \) \( \newcommand{\norm}[1]{\|{#1}\|} \) \( \newcommand{\arccot}{ \, \mathrm{arccot} \, } \) \( \newcommand{\arcsec}{ \, \mathrm{arcsec} \, } \) \( \newcommand{\arccsc}{ \, \mathrm{arccsc} \, } \) \( \newcommand{\sech}{ \, \mathrm{sech} \, } \) \( \newcommand{\csch}{ \, \mathrm{csch} \, } \) \( \newcommand{\arcsinh}{ \, \mathrm{arcsinh} \, } \) \( \newcommand{\arccosh}{ \, \mathrm{arccosh} \, } \) \( \newcommand{\arctanh}{ \, \mathrm{arctanh} \, } \) \( \newcommand{\arccoth}{ \, \mathrm{arccoth} \, } \) \( \newcommand{\arcsech}{ \, \mathrm{arcsech} \, } \) \( \newcommand{\arccsch}{ \, \mathrm{arccsch} \, } \)

17Calculus Precalculus - Solve Logarithms

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Solving Equations Involving Logarithms

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When we have a variable in an exponent, we need to move it out of the exponent to determine it's value. To do that, we use the laws listed on the laws of logarithm page. Also, if the variable is inside a logarithm, we use exponentials to get it out of the logarithm. Here is how it works.

Next

Okay, after working these practice problems, you will be ready to tackle some application problems involving exponentials and logarithms, starting with exponential growth and decay.

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Practice

Unless otherwise instructed, solve these problems using the exponential \(e\), giving your answers in exact terms.

Basic

\( \log_2(x+1) + \log_2(5x+1) = 6 \)

Problem Statement

Solve \( \log_2(x+1) + \log_2(5x+1) = 6 \)

Solution

3032 video solution

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\( \ln x = 7 \)

Problem Statement

Solve \( \ln x = 7 \)

Solution

3046 video solution

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\( \log_3(10x+1) - \log_3(x+1) = 2 \)

Problem Statement

Solve \( \log_3(10x+1) - \log_3(x+1) = 2 \)

Solution

3034 video solution

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\( \log_2 16 = x \)

Problem Statement

Solve \( \log_2 16 = x \)

Solution

3041 video solution

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\( \log_x 81 = 4 \)

Problem Statement

Solve \( \log_x 81 = 4 \)

Solution

3042 video solution

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\( \log_{32} x = 4/5 \)

Problem Statement

Solve \( \log_{32} x = 4/5 \)

Solution

3043 video solution

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\( \log_3(5x+1) = 4 \)

Problem Statement

Solve \( \log_3(5x+1) = 4 \)

Solution

3044 video solution

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\( \log x = 24 \)

Problem Statement

Solve \( \log x = 24 \)

Solution

3045 video solution

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\( \log_7 (x^2+3x+9) = 2 \)

Problem Statement

Solve \( \log_7 (x^2+3x+9) = 2 \)

Solution

3047 video solution

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\( \ln(3x-2) = 5 \)

Problem Statement

Solve \( \ln(3x-2) = 5 \)

Solution

3048 video solution

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\( 4\ln(2x-1) + 3 = 11 \)

Problem Statement

Solve \( 4\ln(2x-1) + 3 = 11 \)

Solution

3049 video solution

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\( \log_3 (5x+2) = \log_3 (7x-8) \)

Problem Statement

Solve \( \log_3 (5x+2) = \log_3 (7x-8) \)

Solution

3050 video solution

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\( \log_2 (x^2+4x) = \log_2 (5) \)

Problem Statement

Solve \( \log_2 (x^2+4x) = \log_2 (5) \)

Solution

3051 video solution

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\( \log_2 x + \log_2 (x+4) = 5 \)

Problem Statement

Solve \( \log_2 x + \log_2 (x+4) = 5 \)

Solution

3052 video solution

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\( \log_3 (x+1) = 3 - \log_3 (x=7) \)

Problem Statement

Solve \( \log_3 (x+1) = 3 - \log_3 (x=7) \)

Solution

3053 video solution

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\( \log_4 (2x+6) - \log_4 (x-1) = 1 \)

Problem Statement

Solve \( \log_4 (2x+6) - \log_4 (x-1) = 1 \)

Solution

3054 video solution

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\( \log_2 (x+3) = 4 + \log_2 (x-3) \)

Problem Statement

Solve \( \log_2 (x+3) = 4 + \log_2 (x-3) \)

Solution

3055 video solution

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Advanced

\( \log x^{\log x} = 49 \)

Problem Statement

Solve \( \log x^{\log x} = 49 \)

Solution

3056 video solution

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\( \log x^2 = (\log x)^2 \)

Problem Statement

Solve \( \log x^2 = (\log x)^2 \)

Solution

3057 video solution

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\( \log (\log x) = 4 \)

Problem Statement

Solve \( \log (\log x) = 4 \)

Solution

3058 video solution

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Topics You Need To Understand For This Page

basics of logarithms

laws of logarithms

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Practice Instructions

Unless otherwise instructed, solve these problems using the exponential \(e\), giving your answers in exact terms.

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