Logarithm equations

Question

It has been years since I had to take a math course. I am currently in an advanced Computer Science course and I have been hit with a hw assignment that is bringing back all the math I used to crank out.

Can someone help me understand this question?

Show that $\log_a x = c \log_b x$ for some constant c (expressed only in terms of the constants a and b).

Answer 1

If I understood your question correctly:

$$ \log_a x = c\log_b x \ \implies \ c = \frac{\log_a x}{\log_b x}$$

By the law of logarithm, $\log_a b = \frac{1}{\log_b a}$

Therefore,

$$ c = \frac{\log_x b}{\log_x a}= \log_a b$$

In the last line, I used the following result (change of base to x):

$$ \log_k h = \frac{\log_p h}{\log_p k}$$

Answer 2

Hint:

$\log_a x = \frac{\ln x}{\ln a}$, for any $x,a>0$ ($a\neq 1$).

Answer 3

$$\log_a{x}=c\log_b{x}$$

But $$\log_a{x}=\frac{\ln x}{\ln a}$$ so $$\frac{\ln x}{\ln a}=c\frac{\ln x}{\ln b}$$ Therefore, $$\frac{\ln b}{\ln a}=c\frac{\ln x}{\ln x}= c(1)$$

Your final result is: $$c=\frac{\ln b}{\ln a}$$