# How do you solve 3^b=17?

Mar 15, 2018

#### Answer:

$b = 2.5789$

#### Explanation:

Lets take the logarithm of both sides of the equation:

$b \cdot \log 3 = \log 17$

then divide both sides by $\log 3$:

$b = \log \frac{17}{\log} 3$

My pocket calculator (HP 15C) reads

$b = 2.5789$

Mar 15, 2018

#### Answer:

Real solution:

$b = \ln \frac{17}{\ln} 3$

Complex solutions:

$b = \frac{\ln 17 + 2 k \pi i}{\ln} 3 \text{ }$ for any integer $k$

#### Explanation:

Given:

${3}^{b} = 17$

Note that ${e}^{2 k \pi i} = 1$ for any integer $k$.

So, if ${e}^{a} = b$ then $a = \ln b + 2 k \pi i$ for any integer $k$.

So while we find the real solution by taking the real valued natural log, we can also add any integer multiple of $2 \pi i$ to find all the complex solutions too...

Take natural log of both sides of the given equation to get:

$b \ln 3 = \ln 17 \textcolor{g r e y}{+ 2 k \pi i}$

Divide both sides by $\ln 3$ to get:

$b = \frac{\ln 17 \textcolor{g r e y}{+ 2 k \pi i}}{\ln} 3$