# Thread: Logarithmic equation is equivalent to exponential equations

1. ## Logarithmic equation is equivalent to exponential equations

Hello,

I was wondering how to change a logarithmic equation to a exponential equation.
For example: 9x = log6​ d

2. ## Re: Logarithmic equation is equivalent to exponential equations

Originally Posted by IStandAlone99
Hello,

I was wondering how to change a logarithmic equation to a exponential equation.
For example: 9x = log6​ d
in this case, since the logarithm is base 6, you would get

$$9x=\log_6(d) \Rightarrow 6^{9x}=d$$

alternatively you could convert to natural logs

$$9x=\frac{\ln(d)}{\ln(6)}$$

$$\Large e^{9x}=e^{\frac{\ln(d)}{\ln(6)}}=d^{\frac{1}{\ln(6 )}}$$

3. ## Re: Logarithmic equation is equivalent to exponential equations

Originally Posted by IStandAlone99
Hello,

I was wondering how to change a logarithmic equation to a exponential equation.
For example: 9x = log6​ d
Here is the general rule

$Given\ b,\ c > 0\ and\ b \ne 1: a = log_b(c) \iff c = b^a.$ This is a definition and needs to be memorized.

So, as Romsek has already explained, $9x = log_6(d) \implies d = 6^{9x}.$

The only tricky thing is that sometimes the base of the logarithm is implied.