what is the real value of x in the equation log² 24 – log³ 3 = log5 x

I am very confused with this question :/

Not clear on your notation.

Does log² 24 mean (log 24)^2 or did you mean log2 ?

same question about log³ 3 and log5 x

I understand that this equation involving logarithms can be confusing. To find the real value of x in the equation log² 24 - log³ 3 = log5 x, we need to simplify the logarithmic expressions and solve for x.

Let's break down the equation step by step:

Step 1: Simplify the logarithmic expressions

To simplify log² 24, we can rewrite it as log2(24). Similarly, log³ 3 can be rewritten as log3(3). Therefore, the equation becomes:

log2(24) - log3(3) = log5(x)

Step 2: Evaluate the logarithmic expressions

Logarithms can be evaluated using the logarithmic properties and change of base formula.

Since log2(24) means "the exponent to which we raise 2 to get 24," we need to find the value of x that satisfies this equation.

To find the value of log2(24), we can use the change of base formula, which states that logb(x) = logc(x) / logc(b), where b, c, and x are positive real numbers. Applying this formula, we can rewrite log2(24) as:

log2(24) = log10(24) / log10(2)

Similarly, log3(3) can be rewritten as:

log3(3) = log10(3) / log10(3)

Therefore, the equation becomes:

log10(24) / log10(2) - log10(3) / log10(3) = log5(x)

Step 3: Combine and simplify the terms on the left side

To combine the terms on the left side, we need to have a common denominator. Since log10(2) and log10(3) are both logarithms with base 10, the common denominator is log10(2) * log10(3).

So, the equation becomes:

(log10(24) * log10(3) - log10(3) * log10(2)) / (log10(2) * log10(3)) = log5(x)

Simplifying the numerator further, we have:

(log10(24 * 3) - log10(3 * 2)) / (log10(2) * log10(3)) = log5(x)

(log10(72) - log10(6)) / (log10(2) * log10(3)) = log5(x)

Now, we can subtract the logarithms in the numerator:

log10(72/6) / (log10(2) * log10(3)) = log5(x)

log10(12) / (log10(2) * log10(3)) = log5(x)

Step 4: Solve for x

To solve for x, we can rewrite the equation in exponential form using the definition of logarithms. In exponential form, logb(x) = y is equivalent to by = x.

Therefore, the equation becomes:

10^(log10(12) / (log10(2) * log10(3))) = x

Now, we can evaluate the right side of the equation using a calculator:

x ≈ 5.913

So, the approximate real value of x in the equation log² 24 - log³ 3 = log5 x is approximately 5.913.

Please note that rounding may be necessary based on the requirements of your specific question.