Test 21 (Lessons 41–42): Common and Natural Logs Solutions

Approximate the value to four decimal places.

$e^{- 4} + \ln 2$

0.7115

$\frac{6}{2 \log 8}$

3.3219

Write the expression with a common log using the Change of Base Rule.

$\log_{m} {(11)}^{2}$

$\frac{2 \log 11}{\log m}$

$\log_{27} M$

$\frac{\log M}{\log 27}$

Write the equation $e^{4 x} = 126$ as a natural log. Do not solve.

$\ln 126 = 4 x$

Solve using common logarithms.

$5^{x} = 21$

$\begin{array}{rcl} \log 5^{x} & = & \log 21 \\ x \log 5 & = & \log 21 \\ x & = & \frac{\log 21}{\log 5} \end{array}$

$8^{(2 x + 1)} = 15$

$\begin{array}{rcl} \log 8^{(2 x + 1)} & = & \log 15 \\ (2 x + 1) \log 8 & = & \log 15 \\ 2 x + 1 & = & \frac{\log 15}{\log 8} \\ 2 x & = & \frac{\log 15}{\log 8} - 1 \\ x & = & \frac{\log 15}{2 \log 8} - \frac{1}{2} \end{array}$

Write $\log 135$ in terms of X and Y when $\log 5 = X$ and $\log 3 = Y .$

$\begin{array}{rcl} \log 135 & = & \log (3^{3} \cdot 5) \\ = & \log 3^{3} + \log 5 \\ = & 3 \log 3 + \log 5 \\ = & 3 (Y) + (X) \\ = & X + 3 Y \end{array}$

Evaluate.

$e^{\ln 9 + \ln 0.5}$

$e^{\ln (9 \cdot 0.5)} e^{\ln 4.5} 4.5$

${(\ln e^{4})}^{- 1}$

${(4)}^{- 1} \frac{1}{4}$

Solve using natural logarithms.

$4^{x + 5} = 1$

$\begin{array}{rcl} \ln 4^{x + 5} & = & \ln 1 \\ (x + 5) \ln 4 & = & \ln 1 \\ x + 5 & = & \frac{\ln 1}{\ln 4} \\ x & = & \frac{\ln 1}{\ln 4} - 5 \\ x & = & - 5 \end{array}$

$\frac{1}{2} \ln (x + 3) = \ln 5$

$\begin{array}{rcl} \ln {(x + 3)}^{\frac{1}{2}} & = & \ln 5 \\ {({(x + 3)}^{\frac{1}{2}})}^{2} & = & {(5)}^{2} \\ x + 3 & = & 25 \\ x & = & 22 \end{array}$