Use properties of logarithms to find the exact value of the expression
log_7 (343 * 7^15)
log7 (343*7^15)
= log7 343 + log7 7^15
= log7 7^3 + 15log77
= 3log77 + 15log77
= 3(1) + 15(1)
= 18
To find the exact value of the expression `log_7(343 * 7^15)`, we can use the properties of logarithms.
The first property we'll use is the product rule, which states that the logarithm of a product is equal to the sum of the logarithms of the individual factors. Mathematically, this property can be expressed as log_b(x * y) = log_b(x) + log_b(y).
Next, we can simplify the expression by applying the power rule, which states that the logarithm of a number raised to a power is equal to the product of the power and the logarithm of the number. In other words, log_b(x^y) = y * log_b(x).
Now, let's apply these properties step by step:
Step 1: Simplify the expression inside the logarithm:
First, we have 343 * 7^15. Since 343 is equal to 7^3, we can rewrite the expression as (7^3) * (7^15).
Step 2: Apply the product rule:
Using the product rule, we can rewrite the expression as log_7(7^3) + log_7(7^15).
Step 3: Apply the power rule:
Using the power rule, we can further simplify the expression to 3 + 15 * log_7(7).
Step 4: Simplify further:
The logarithm of any base with its own base is equal to 1. So, log_7(7) = 1. Therefore, we can simplify the expression to 3 + 15 * 1, which is equal to 3 + 15.
Step 5: Calculate the final result:
Adding 3 and 15 gives us the final answer of 18.
Therefore, the exact value of the expression log_7(343 * 7^15) is 18.