exponential functions.

Question

I am confused of solving exponential functions they look easy but can't solve it. 1: $$\large e^{8\cdot\ln(b^{1/4})}$$

and this one solving for x: 1: $$\ln(6x-2) = 5$$

FYI : It's not an assignment. i am trying to solve it for my understanding

Answer 1

For the first one, you probably just need to simplify it. Because it isn't an equation, you can't really "solve'' it for anything.

Here's how you could simplify that expression:

\begin{align*} e^{8\ln\left(b^{\frac{1}{4}}\right)} &= e^{\ln\left(b^{\frac{1}{4}}\right)^8} \\ &= e^{\ln\left(b^{\frac{8}{4}}\right)} \\ &= e^{\ln\left(b^{2}\right)} \\ &= b^2 \end{align*} The first step involves using exponent rules for natural logs (namely, that $a\log b = \log b^a$), and the last part uses the fact that $e^x$ and $\ln x$ are $\textit{inverses}$; i.e., that $e^{\ln x} = x$ and $\ln \left(e^x\right) = x$.

Answer 2

The first one isn't equal to anything to solve so it's unknown, the other is $$\ln(6x-2)=5$$ $$6x-2=e^5$$ $$6x=e^5+2$$ $$x=\frac{e^5+2}{6}$$