Solving Exponential Equations

Question 1 of 5

Solve for

x

$x$ using Change of Base

3^{x + 1} = 8

$3^(x+1)=8$

Round answer to

4

$4$ decimal places

$x =$ $x=$ (0.8928)

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Exponent Form

N = a^{x}

$\color{#00880a}{N}={\color{#9a00c7}{a}}^x$

Logarithmic Form

x = {log}_{a} N

$x=\log_{\color{#9a00c7}{a}} \color{#00880a}{N}$

Change of Base Formula

{log}_{a} N = \frac{{log}_{b} N}{{log}_{b} a}

$\log_\color{#9a00c7}{a} \color{#00880A}{N}=\frac{\log_b \color{#00880A}{N}}{\log_b \color{#9a00c7}{a}}$

Transform the given exponential equation to logarithmic form

$\color{#9a00c7}{3}^{x+1}$	$=$ $=$	$\color{#00880a}{8}$
$x+1$	$=$ $=$	$\log_\color{#9a00c7}{3} \color{#00880a}{8}$

Use the change of base formula, then use the calculator to solve the logarithm

$x+1$	$=$ $=$	$\log_\color{#9a00c7}{3} \color{#00880a}{8}$

$x+1$	$=$ $=$	$\frac{\log_{10} \color{#00880a}{8}}{\log_{10} \color{#9a00c7}{3}}$	Calculators use $10$ $10$ as base for the log function

$x+1$	$=$ $=$	$1.8928$	Compute using the calculator
$x + 1$ $x+1$ $- 1$ $-1$	$=$ $=$	$1.8928$ $1.8928$ $- 1$ $-1$	Subtract $1$ $1$ from both sides
$x$ $x$	$=$ $=$	$0.8928$ $0.8928$

x = 0.8928

$x=0.8928$

Question 2 of 5

Solve for

x

$x$

2^{x} = 5^{x - 1}

$2^x=5^(x-1)$

Round answer to

4

$4$ decimal places

$x =$ $x=$ (1.7565)

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Laws of Logarithms

{log}_{b} x^{p} = p {log}_{b} x

$\log_b x^\color{#004ec4}{p}=\color{#004ec4}{p}\log_b x$

Insert logarithms of the same base to both sides, then solve for

x

$x$

$\color{#D800AD}{2^x}$	$=$ $=$	$\color{#D800AD}{5^{x-1}}$
$\log_{10} \color{#D800AD}{2^x}$	$=$ $=$	$\log_{10} \color{#D800AD}{5^{x-1}}$	Calculators use base $10$ $10$ for log function
$\color{#004ec4}{x}\log_{10} 2$	$=$ $=$	$\color{#004ec4}{(x-1)}\log_{10} 5$	${log}_{b} x^{p} = p {log}_{b} x$ $log_b x^p=p log_b x$
$x {log}_{10} 2$ $xlog_(10) 2$	$=$ $=$	$x {log}_{10} 5 - {log}_{10} 5$ $x log_(10) 5-log_(10) 5$	Distribute
$x {log}_{10} 2$ $xlog_(10) 2$ $+ {log}_{10} 5$ $+log_(10) 5$	$=$ $=$	$x {log}_{10} 5 - {log}_{10} 5$ $x log_(10) 5-log_(10) 5$ $+ {log}_{10} 5$ $+log_(10) 5$	Add ${log}_{10} 5$ $log_(10) 5$ to both sides
$x {log}_{10} 2 + {log}_{10} 5$ $xlog_(10) 2+log_(10) 5$	$=$ $=$	$x {log}_{10} 5$ $x log_(10) 5$
$x {log}_{10} 2 + {log}_{10} 5$ $xlog_(10) 2+log_(10) 5$ $- x {log}_{10} 2$ $-xlog_(10) 2$	$=$ $=$	$x {log}_{10} 5$ $x log_(10) 5$ $- x {log}_{10} 2$ $-xlog_(10) 2$	Subtract $x {log}_{10} 2$ $xlog_(10) 2$ from both sides
${log}_{10} 5$ $log_(10) 5$	$=$ $=$	$x ({log}_{10} 5 - {log}_{10} 2)$ $x(log_(10) 5-log_(10) 2)$	Factorize

$\frac{\log_{10} 5}{\color{#CC0000}{\log_{10} 5-\log_{10} 2}}$	$=$ $=$	$\frac{x(\log_{10} 5-\log_{10} 2)}{\color{#CC0000}{\log_{10} 5-\log_{10} 2}}$	Divide both sides by ${log}_{10} 5 - {log}_{10} 2$ $log_(10) 5-log_(10) 2$

$\frac{{log}_{10} 5}{{log}_{10} 5 - {log}_{10} 2}$ $(log_(10) 5)/(log_(10) 5-log_(10) 2)$	$=$ $=$	$x$ $x$

$x$ $x$	$=$ $=$	$1.7565$ $1.7565$	Compute using the calculator

x = 1.7565

$x=1.7565$

Question 3 of 5

Solve for

x

$x$

10^{3 x - 2} = 5^{4 x}

$10^(3x-2)=5^(4x)$

Round answer to

3

$3$ decimal places

$x =$ $x=$ (9.798)

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Laws of Logarithms

{log}_{b} x^{p} = p {log}_{b} x

$\log_b x^\color{#004ec4}{p}=\color{#004ec4}{p}\log_b x$

{log}_{b} b = 1

$\log_{\color{#9a00c7}{b}} \color{#9a00c7}{b}=1$

Insert logarithms of the same base to both sides, then solve for

x

$x$

$\color{#D800AD}{10^{3x-2}}$	$=$ $=$	$\color{#D800AD}{5^{4x}}$
$\log_{10} \color{#D800AD}{10^{3x-2}}$	$=$ $=$	$\log_{10} \color{#D800AD}{5^{4x}}$	Calculators use base $10$ $10$ for log function
$\color{#004ec4}{(3x-2)}\log_{10} 10$	$=$ $=$	$\color{#004ec4}{(4x)}\log_{10} 5$	${log}_{b} x^{p} = p {log}_{b} x$ $log_b x^p=p log_b x$
$(3x-2)(\color{#9a00c7}{1})$	$=$ $=$	$\color{#004ec4}{(4x)}\log_{10} 5$	${log}_{b} b = 1$ $log_b b=1$
$3 x - 2$ $3x-2$ $+ 2$ $+2$	$=$ $=$	$4 x {log}_{10} 5$ $4x log_(10) 5$ $+ 2$ $+2$	Add $2$ $2$ to both sides
$3 x$ $3x$	$=$ $=$	$4 x {log}_{10} 5 + 2$ $4x log_(10) 5+2$
$3 x$ $3x$ $- 4 x {log}_{10} 5$ $-4xlog_(10) 5$	$=$ $=$	$4 x {log}_{10} 5 + 2$ $4x log_(10) 5+2$ $- 4 x {log}_{10} 5$ $-4xlog_(10) 5$	Subtract $4 x {log}_{10} 5$ $4xlog_(10) 5$ from both sides
$x (3 - 4 {log}_{10} 5)$ $x(3-4log_(10) 5)$	$=$ $=$	$2$ $2$	Factorize

$\frac{x(3-4\log_{10} 5)}{\color{#CC0000}{3-4\log_{10} 5}}$	$=$ $=$	$\frac{2}{\color{#CC0000}{3-4\log_{10} 5}}$	Divide both sides by $3 - 4 {log}_{10} 5$ $3-4\log_(10) 5$

$x$ $x$	$=$ $=$	$\frac{2}{3 - 4 {log}_{10} 5}$ $2/(3-4\log_(10) 5)$

$x$ $x$	$=$ $=$	$9.798$ $9.798$	Compute using the calculator

x = 9.798

$x=9.798$

Question 4 of 5

Solve for

x

$x$

6^{x} = 0.00025

$6^x=0.00025$

Round answer to

3

$3$ decimal places

$x =$ $x=$ (-4.629)

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Laws of Logarithms

{log}_{b} x^{p} = p {log}_{b} x

$\log_b x^\color{#004ec4}{p}=\color{#004ec4}{p}\log_b x$

Transform the decimal to exponential form

$6^x$	$=$ $=$	$0.00025$
$6^x$	$=$ $=$	$\frac{25}{100 000}$	$0.00025 = \frac{25}{100000}$ $0.00025=25/(100 000)$

$6^x$	$=$ $=$	$\frac{1}{4000}$	Simplify

$6^x$	$=$ $=$	$4000^{-1}$	Reciprocate $\frac{1}{4000}$ $1/(4000)$

Insert logarithms of the same base to both sides, then solve for

x

$x$

$\color{#D800AD}{6^{x}}$	$=$ $=$	$\color{#D800AD}{4000^{-1}}$
$\log_{10} \color{#D800AD}{6^{x}}$	$=$ $=$	$\log_{10} \color{#D800AD}{4000^{-1}}$	Calculators use $10$ $10$ as base for the log function
$\color{#004ec4}{x}\log_{10} 6$	$=$ $=$	$\color{#004ec4}{(-1)}\log_{10} {4000}$	${log}_{b} x^{p} = p {log}_{b} x$ $log_b x^p=p log_b x$

$\frac{x\log_{10} 6}{\color{#CC0000}{\log_{10} 6}}$	$=$ $=$	$\frac{(-1)\log_{10} 4000}{\color{#CC0000}{\log_{10} 6}}$	Divide both sides by ${log}_{10} 6$ $log_(10) 6$

$x$ $x$	$=$ $=$	$\frac{(- 1) {log}_{10} 4000}{{log}_{10} 6}$ $((-1)log_(10) 4000)/(log_(10) 6)$

$x$ $x$	$=$ $=$	$- 4.629$ $-4.629$	Compute using calculator

x = - 4.629

$x=-4.629$

Question 5 of 5

Solve for

x

$x$

{(\frac{3}{5})}^{x} = 10^{- 5}

$(3/5)^x=10^(-5)$

Round answer to

4

$4$ decimal places

$x =$ $x=$ (-22.5379)

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Laws of Logarithms

{log}_{b} x^{p} = p {log}_{b} x

$\log_b x^\color{#004ec4}{p}=\color{#004ec4}{p}\log_b x$

{log}_{b} b = 1

$\log_{\color{#9a00c7}{b}} \color{#9a00c7}{b}=1$

Insert logarithms of the same base to both sides, then solve for

x

$x$

$\color{#D800AD}{\frac{3}{5}^{x}}$	$=$ $=$	$\color{#D800AD}{10^{-5}}$

$\log_{10} \color{#D800AD}{\frac{3}{5}^{x}}$	$=$ $=$	$\log_{10} \color{#D800AD}{10^{-5}}$	Calculators use $10$ $10$ as base for the log function

$\color{#004ec4}{x}\log_{10} \frac{3}{5}$	$=$ $=$	$\color{#004ec4}{(-5)}\log_{10} 10$	${log}_{b} x^{p} = p {log}_{b} x$ $log_b x^p=p log_b x$

$x\log_{10} \frac{3}{5}$	$=$ $=$	$(-5)(\color{#9a00c7}{1})$	${log}_{b} b = 1$ $log_b b=1$

$\frac{x\log_{10} \frac{3}{5}}{\color{#CC0000}{\log_{10} \frac{3}{5}}}$	$=$ $=$	$\frac{-5}{\color{#CC0000}{\log_{10} \frac{3}{5}}}$	Divide both sides by ${log}_{10} (\frac{3}{5})$ $\log_(10) (3/5)$

$x$ $x$	$=$ $=$	$- \frac{5}{{log}_{10} (\frac{3}{5})}$ $-5/(\log_(10) (3/5))$

$x$ $x$	$=$ $=$	$- 22.5379$ $-22.5379$	Compute using the calculator

x = - 22.5379

$x=-22.5379$

Solving Exponential Equations

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Quiz summary

Information

1. Question

Hint

Great Work!

Exponent Form

Logarithmic Form

Change of Base Formula

2. Question

Hint

Correct!

Laws of Logarithms

3. Question

Hint

Keep Going!

Laws of Logarithms

4. Question

Hint

Fantastic!

Laws of Logarithms

5. Question

Hint

Excellent!

Laws of Logarithms

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