Question
The logarithmic equation
y= logb x
x passes through the point
(1/7,−1)
(
7
1
, −1) . What is the value of b?
b =
y= logb x
x passes through the point
(1/7,−1)
(
7
1
, −1) . What is the value of b?
b =
Answers
GPT-4o mini
To determine the value of \( b \) in the equation \( y = \log_b x \), given that the point \( \left( \frac{1}{7}, -1 \right) \) lies on the graph of the logarithmic function, we can substitute the coordinates of the point into the equation.
The equation is:
\[
y = \log_b x
\]
Substituting the point \( \left( \frac{1}{7}, -1 \right) \):
\[
-1 = \log_b \left( \frac{1}{7} \right)
\]
From the definition of logarithms, we have:
\[
\log_b a = c \implies a = b^c
\]
Applying that here, we can rewrite our logarithmic equation:
\[
\frac{1}{7} = b^{-1}
\]
This implies:
\[
b^{-1} = \frac{1}{7}
\]
Taking the reciprocal of both sides gives:
\[
b = 7
\]
Thus, the value of \( b \) is:
\[
\boxed{7}
\]
The equation is:
\[
y = \log_b x
\]
Substituting the point \( \left( \frac{1}{7}, -1 \right) \):
\[
-1 = \log_b \left( \frac{1}{7} \right)
\]
From the definition of logarithms, we have:
\[
\log_b a = c \implies a = b^c
\]
Applying that here, we can rewrite our logarithmic equation:
\[
\frac{1}{7} = b^{-1}
\]
This implies:
\[
b^{-1} = \frac{1}{7}
\]
Taking the reciprocal of both sides gives:
\[
b = 7
\]
Thus, the value of \( b \) is:
\[
\boxed{7}
\]