Calculate log₁₅(9) + log₁₅(25): A Step-by-Step Solution

This tutorial demonstrates how to calculate the exact value of the logarithmic expression log₁₅(9) + log₁₅(25). We will use the product rule of logarithms to simplify and solve the expression.

1. Apply the Product Rule:

The product rule of logarithms states that the sum of two logarithms with the same base is equal to the logarithm of the product of their arguments. Applying this rule, we get:

log₁₅(9) + log₁₅(25) = log₁₅(9 * 25)

2. Simplify the Expression:

Simplifying the multiplication within the logarithm gives:

log₁₅(9 * 25) = log₁₅(225)

3. Find the Exponent:

Now, we need to find the exponent to which 15 must be raised to obtain 225. In other words, we need to solve the equation:

15^x = 225

4. Express 225 as a Power of 15:

We can express 225 as a power of 15:

225 = 15²

5. Solve for x:

Substituting this back into the equation, we get:

15^x = 15²

Since the bases are the same, the exponents must be equal. Therefore:

x = 2

Solution:

The exact value of the expression log₁₅(9) + log₁₅(25) is 2.