How do you solve this math problem?

Question

What is the real value of x in the equation:
log of 24 in base 2 - log of 3 in base 2 = log of x in base 5 ?


What are the steps to solving this problem? Thanks in advance.

Answer 1

For this problem we need to use a few properties, first of all is the Quotient Property:

log of 24 in base 2 - log of 3 in base 2

= log of 24/3 in base 2

= log of 8 in base 2 = log of x in base 5

Next we need to use the Change of base property, which says:

log of a in base b = log of b in base 10 / log of a in base 10

log of 8 in base 10 / log of 2 in base 10 = log of x in base 5

.903 / .301 = log of x in base 5

3 = log of x in base 5

Now we transform the equation into an exponential one to solve for x, we will use the following format:

log of x in base 10 is the same as saying

10^x = x

3 = log of x in base 5

5^3 = x

125 = x

And there you go.

Answer 2

First calculate log of 24 in base 2 - log of 3 in base 2

log of 24 in base 2 - log of 3 in base 2 =
= log of 3 * 2^3 in base 2 - log of 3 in base 2 =
log (base 2) 3 + log (base 2) 2^3 - log (base 2) 3 =
= log (base 2) 2^3 = 3

log of 24 in base 2 - log of 3 in base 2 = log of x in base 5 ==>
3 = log of x in base 5 // 5^(each side)
5^3 = x

x=125

Answer 3

You could convert bases as follows:
If y = logn (x), ie base n
then n^y = x
Suppose we want to convert to base m
Take logm of both sides
logm (n^y) = y*logm(n) = logm(x)
y = logm(x)/logm(n)

So log2(24) = log10(24)/log10(2) = 4.585
log2(3) = log10(3)/log10(2) = 1.585
log5(x) = log10(x)/log10(5)

log10(x) = 3*log10(5) = log10(5^3)
x = 5^3 = 125