Change Of Base Formula

Question 1 of 4

Derive the Change of Base formula from the general equation

x = {log}_{a} N

$x=log_a N$

1. ${log}_{a} N = ({log}_{b} N) ({log}_{b} a)$ $log_a N=(log_b N)(log_b a)$
2. ${log}_{a} N = \frac{{log}_{a} N}{{log}_{a} b}$ $log_a N=(log_a N)/(log_a b)$
3. ${log}_{a} N = \frac{{log}_{b} a}{{log}_{b} N}$ $log_a N=(log_b a)/(log_b N)$
4. ${log}_{a} N = \frac{{log}_{b} N}{{log}_{b} a}$ $log_a N=(log_b N)/(log_b a)$

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Exponent Form

N = a^{x}

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

Logarithmic Form

x = {log}_{a} N

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

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 general logarithmic equation to exponent form

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

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

x

$x$

$\color{#D800AD}{N}$	$=$ $=$	$\color{#D800AD}{a^x}$
$\log_b \color{#D800AD}{N}$	$=$	$\log_b \color{#D800AD}{a^x}$
$\log_b N$	$=$	$\log_b a^\color{#004ec4}{x}$
$\log_b N$	$=$	$\color{#004ec4}{x}\log_b a$	$log_b x^p=p log_b x$
$log_b N$ $divide log_b a$	$=$	$x log_b a$ $divide log_b a$	Divide both sides by $log_b a$

$(log_b N)/(log_b a)$	$=$	$x$

$x$	$=$	$(log_b N)/(log_b a)$

Also, remember that,

$x=log_a N$

Hence, the Change of Base formula is:

$log_a N=(log_b N)/(log_b a)$

Question 2 of 4

Evaluate using Change of Base

$log_2 9$

Round answer to

$5$ decimal places

$log_2 9=$ (3.16993)

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Exponent Form

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

Logarithmic Form

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

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

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

$\log_\color{#9a00c7}{2} \color{#00880a}{9}$	$=$	$\frac{\log_{10} \color{#00880a}{9}}{\log_{10} \color{#9a00c7}{2}}$	Calculators use $10$ as base for the log function

	$=$	$3.16993$	Compute using the calculator

$log_2 9=3.16993$

Question 3 of 4

Solve for

$x$ using Change of Base

$5^x=11$

Round answer to

$4$ decimal places

$x=$ (1.4899)

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Exponent Form

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

Logarithmic Form

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

Change of Base Formula

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

Transform the given exponential equation to logarithmic form

$\color{#9a00c7}{5}^x$	$=$	$\color{#00880a}{11}$
$x$	$=$	$\log_\color{#9a00c7}{5} \color{#00880a}{11}$

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

$x$	$=$	$\log_\color{#9a00c7}{5} \color{#00880a}{11}$

$x$	$=$	$\frac{\log_{10} \color{#00880a}{11}}{\log_{10} \color{#9a00c7}{5}}$	Calculators use $10$ as base for the log function

$x$	$=$	$1.4899$	Compute using the calculator

$x=1.4899$

Question 4 of 4

Solve for

$x$ using Change of Base

$2^x=0.062$

Round answer to

$4$ decimal places

$x=$ (-4.0116)

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Exponent Form

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

Logarithmic Form

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

Change of Base Formula

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

Transform the given exponential equation to logarithmic form

$\color{#9a00c7}{2}^x$	$=$	$\color{#00880a}{0.062}$
$x$	$=$	$\log_\color{#9a00c7}{2} \color{#00880a}{0.062}$

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

$x$	$=$	$\log_\color{#9a00c7}{2} \color{#00880a}{0.062}$

$x$	$=$	$\frac{\log_{10} \color{#00880a}{0.062}}{\log_{10} \color{#9a00c7}{2}}$	Calculators use $10$ as base for the log function

$x$	$=$	$-4.0116$	Compute using the calculator

$x=-4.0116$

Change Of Base Formula

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Quiz summary

Information

1. Question

Hint

Well Done!

Exponent Form

Logarithmic Form

Laws of Logarithms

2. Question

Hint

Nice Job!

Exponent Form

Logarithmic Form

Change of Base Formula

3. Question

Hint

Excellent!

Exponent Form

Logarithmic Form

Change of Base Formula

4. Question

Hint

Fantastic!

Exponent Form

Logarithmic Form

Change of Base Formula

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