If log_5(25) = x, then what is the value of log_5(125) in terms of x?

If log_5(25) = x, then what is the value of log_5(125) in terms of x?

Correct Answer: 3

Step 1: Understand that log_5(25) = x means that 5 raised to the power of x equals 25.
Step 2: Rewrite 25 as a power of 5. Since 25 = 5^2, we have 5^x = 5^2.
Step 3: From the equation 5^x = 5^2, we can conclude that x = 2.
Step 4: Now, we need to find log_5(125). Rewrite 125 as a power of 5. Since 125 = 5^3, we have log_5(125) = log_5(5^3).
Step 5: Use the property of logarithms that states log_b(a^c) = c * log_b(a). Here, log_5(5^3) = 3 * log_5(5).
Step 6: Since log_5(5) = 1, we have log_5(125) = 3 * 1 = 3.
Step 7: We already found that x = 2, and we need to express log_5(125) in terms of x. Since log_5(125) = 3, we can relate it to x: log_5(125) = (3/2) * x.

Logarithmic Properties – Understanding the properties of logarithms, including the change of base and the power rule.
Exponential Relationships – Recognizing the relationship between logarithmic and exponential forms, particularly with bases and exponents.