log2^24 - log2^3 = log5^x

Question

log2^24 - log2^3 = log5^x

what is X? thanks

Answer 1

To solve the equation log2^24 - log2^3 = log5^x and find the value of x, we will use logarithmic properties and basic algebra.

First, let's rewrite the equation using the logarithmic property: log a^n = n * log a.

log2^24 - log2^3 = log5^x can be rewritten as:
24 * log2 - 3 * log2 = log5^x.

Next, we can combine the logarithms using the property: log a - log b = log(a / b).
24 * log2 - 3 * log2 = log(2^24 / 2^3) = log2^(24-3) = log2^21.

Therefore, the equation simplifies to:
log2^21 = log5^x.

Now, we have two logarithms with different bases (2 and 5). To convert them to the same base, we can use the change of base formula: log a^b = log c^b / log c^a.

Let's convert the equation to a base 10 logarithm:
log2^21 = log5^x can be rewritten as:
log(21) / log(2) = log(5^x) / log(2).

Since both fractions are equal, we can set them equal to each other:
log(21) / log(2) = log(5^x) / log(2).

Now, we can eliminate the common base logarithm log(2) and solve for x:
log(21) = x * log(5).

Divide both sides by log(5):
x = log(21) / log(5).

Using a calculator to evaluate the logarithms, we find:
x ≈ 2.0589.

Therefore, x is approximately 2.0589.

Answer 2