Question
Solve the equation log8(8y)=log8(5y+15) . Remember to check your solution! If there is no solution, put NS in the box. (1 point) y =
Answers
GPT-4o mini
To solve the equation \( \log_8(8y) = \log_8(5y + 15) \), we can use the property that if \( \log_a(x) = \log_a(y) \), then \( x = y \) (as long as the base \( a > 0 \) and \( a \neq 1 \), and \( x, y > 0 \)).
From the given equation, we have:
\[
8y = 5y + 15.
\]
Now, let's solve for \( y \):
1. Subtract \( 5y \) from both sides:
\[
8y - 5y = 15,
\]
which simplifies to:
\[
3y = 15.
\]
2. Now, divide both sides by 3:
\[
y = 5.
\]
Next, we need to check if this solution is valid by substituting \( y = 5 \) back into the original logarithmic expressions to ensure they are defined (i.e., the arguments must be positive):
- For \( \log_8(8y) \):
\[
8y = 8(5) = 40.
\]
- For \( \log_8(5y + 15) \):
\[
5y + 15 = 5(5) + 15 = 25 + 15 = 40.
\]
Both logarithmic expressions are defined since both \( 40 > 0 \).
Thus, the solution to the equation is:
\[
\boxed{5}.
\]
From the given equation, we have:
\[
8y = 5y + 15.
\]
Now, let's solve for \( y \):
1. Subtract \( 5y \) from both sides:
\[
8y - 5y = 15,
\]
which simplifies to:
\[
3y = 15.
\]
2. Now, divide both sides by 3:
\[
y = 5.
\]
Next, we need to check if this solution is valid by substituting \( y = 5 \) back into the original logarithmic expressions to ensure they are defined (i.e., the arguments must be positive):
- For \( \log_8(8y) \):
\[
8y = 8(5) = 40.
\]
- For \( \log_8(5y + 15) \):
\[
5y + 15 = 5(5) + 15 = 25 + 15 = 40.
\]
Both logarithmic expressions are defined since both \( 40 > 0 \).
Thus, the solution to the equation is:
\[
\boxed{5}.
\]