1. logarithmic

Given that , what is the value of ?

Can someone show me how to solve this please? Cheers x

2. Or you can use the facts that $\displaystyle \log_b5^{1/2} = \tfrac12\log_b5$ and $\displaystyle \log_b5^4 = 4\log_b5$. Then the equation simplifies to $\displaystyle \log_b5=4$, which you should be able to solve on a calculator.

3. $\displaystyle -2\log_x(\sqrt{5})+5\log_x(5^4)=76$

Using the change of base theorem

$\displaystyle -2\frac{\frac{1}{2}\ln(5)}{\ln(x)}+20\frac{\ln(5)}{ \ln(x)}=76$

Solving gives

$\displaystyle \ln(x)=\frac{\ln(5)}{4}\implies{x=5^{\frac{1}{4}}}$

4. Originally Posted by Opalg
Or you can use the facts that $\displaystyle \log_b5^{1/2} = \tfrac12\log_b5$ and $\displaystyle \log_b5^4 = 4\log_b5$. Then the equation simplifies to $\displaystyle \log_b5=4$, which you should be able to solve on a calculator.
I deleted my original post in this thread after seeing the other replies thinking I had made an awfull mistake, but looking at these replies I see that they are addressing the problem:

$\displaystyle -2 \log_b(5^{1/2})+5\log_b(5^4)=76$

which seems natural enough because it can be solved in close form.

However that is not my reading of the question, which was to solve:

$\displaystyle -2^{ \log_b(5^{1/2})}+5^{\log_b(5^4)}=76$

which is somewhat different! and I stand by my earlier now deleted post and suggest graphical and/or numerical solution for this.

There is a real solution is close to $\displaystyle b=10.84$.

CB