solve the equation:

log(base2)^(x-3)log(base2)^5 = 2log(base2)^10

I don't know how to solve it, can someone help me? Please and thank you.

your notation makes no sense

on the right side you have

2log(base2)^10
or
2log210

there is no such notation.

The expression on the left is even worse.

this is exactly what my homework sheet says. this is not teacher made, it's from the book. 2log(base2)^10 = log(base2)^10^2 = log(base2)^100

In this and most forums the ^ is used as an exponent indicator

e.g. 2^3 = 23

looking at your last line in the previous reply, I will assume that by
2log(base2)^10 = log(base2)^10^2 = log(base2)^100 you really meant :
2log(base2)10 = log(base2)10^2 = log(base2)100

so your question of
log(base2)^(x-3)log(base2)^5 = 2log(base2)^10 is really
log(base2)(x-3)log(base2)5 = 2log(base2)10 or
log2(x-3)log25 = 2log2100
divide by log25
log2(x-3) = log2100/log25
log2(x-3) = log100/log5 = 2.861353
so x-3 = 2^2.861353
x-3 = 7.26697
x = 10.26697

To solve the equation:

log(base2)^(x-3) * log(base2)^5 = 2 * log(base2)^10

Let's simplify it step by step.

First, we can apply the multiplication rule of logarithms, which states that:

log(base a)^(m) * log(base a)^(n) = log(base a)^(m+n)

Using this rule, we can rewrite the equation as:

log(base 2)^((x-3) * 5) = log(base 2)^(10^2)

Simplifying further, we get:

log(base 2)^(5x-15) = log(base 2)^(100)

Now, since the base of the logarithm is the same, we can equate the two expressions inside the logarithms:

5x - 15 = 100

Next, solve the equation for x.

Add 15 to both sides:

5x = 115

Divide both sides by 5:

x = 23

So, the solution to the equation is:

x = 23