Question

Solve the equation log8 (8y) = log8 (5y+15)

Answer 1

To solve the equation \( \log_8 (8y) = \log_8 (5y + 15) \), we can use the property of logarithms that states if \( \log_a(b) = \log_a(c) \), then \( b = c \) (as long as the logarithm base and arguments are valid).

Thus, we can set the arguments equal to each other:

\[
8y = 5y + 15
\]

Now, we'll solve for \( y \):

1. Subtract \( 5y \) from both sides:

\[
8y - 5y = 15
\]

This simplifies to:

\[
3y = 15
\]

2. Now, divide both sides by 3:

\[
y = 5
\]

Now, let's verify that our solution is valid by substituting \( y = 5 \) back into the original logarithmic expressions.

The expressions are:

- Left side: \( \log_8(8 \cdot 5) = \log_8(40) \)
- Right side: \( \log_8(5 \cdot 5 + 15) = \log_8(25 + 15) = \log_8(40) \)

Both sides are equal:

\[
\log_8(40) = \log_8(40)
\]

Since both sides are equal, the solution is confirmed. Therefore, the solution to the equation is:

\[
\boxed{5}
\]