Solve with Common Logs Solutions

• The Equality Rule for logarithms allows you to rewrite    exponential    equations in terms of    logarithms   .
• Because of the phrase    “if and only if,”    the rule is true forward and backward. 

• This rule determines the    value    of a logarithm when x is    not equal    to a rational number, ℚ. 

• You can write the    exact    answer in terms of logarithms and, if necessary,    approximate    the value with technology.

Example 5

Solve. Write the answer as a logarithm and as a number to four decimal places.

$5^x=12$

Plan

Equality rule for logs

Isolate x

Approximate with technology

$$\begin{gathered} 5^x=12 \\ \log 5^x=\log 12 \end{gathered}$$

Note

If you substitute the log expression rather than the decimal, you will get the exact answer (if you have solved it correctly).


 

Because you are rounding, when the decimal approximation is substituted into the given problem, the result will not be the exact power. However, it will round to the given power if the answer is correct.

Example 6

Solve. Write the answer with common logarithms.

$3^{\frac{x}{5}}=0.4$

Note

Remember, this fraction cannot be simplified because the logarithmic terms are irrational numbers.

Example 7

Solve. Write the answer as a logarithm and as a number to four decimal places.


$3^{x–2}=88$

$\begin{array}{rcl}\log 3^{x–2}& =& \log 88\\ \left(x–2\right)\log 3& =& \log 88\\ x–2& =& \frac{\log 88}{\log 3}\\ x& =& \frac{\log 88}{\log 3}+2\approx 6.0754\end{array}$