Answer :
To determine which equation can be solved using the inverse property, we need to evaluate each option carefully. The inverse property is based on the fact that certain mathematical functions have corresponding inverse functions, which can be used to solve equations.
Let's examine each option one by one:
A. [tex]\(\log_2 x = \log_2 6\)[/tex]
Using the inverse property of logarithms, we can exponentiate both sides with base 2:
[tex]\[ 2^{\log_2 x} = 2^{\log_2 6} \][/tex]
This simplifies to:
[tex]\[ x = 6 \][/tex]
So, option A can be solved using the inverse property of logarithms.
B. [tex]\(e^{3.5} = e^{2x}\)[/tex]
Using the inverse property of the exponential function [tex]\(e\)[/tex], we take the natural logarithm of both sides:
[tex]\[ \ln(e^{3.5}) = \ln(e^{2x}) \][/tex]
This simplifies to:
[tex]\[ 3.5 = 2x \][/tex]
By solving for [tex]\(x\)[/tex], we get:
[tex]\[ x = \frac{3.5}{2} = 1.75 \][/tex]
So, option B can also be solved using the inverse property of the exponential function.
C. [tex]\(10^{2x} = 10^{-6}\)[/tex]
Using the inverse property of the exponential function with a base of 10, we can take the common logarithm (base 10) of both sides:
[tex]\[ \log_{10}(10^{2x}) = \log_{10}(10^{-6}) \][/tex]
This simplifies to:
[tex]\[ 2x = -6 \][/tex]
By solving for [tex]\(x\)[/tex], we get:
[tex]\[ x = -3 \][/tex]
So, option C can be solved using the inverse property of the exponential function with base 10.
D. [tex]\(\log_4 x = -1\)[/tex]
Using the inverse property of logarithms, we can exponentiate both sides with base 4:
[tex]\[ 4^{\log_4 x} = 4^{-1} \][/tex]
This simplifies to:
[tex]\[ x = \frac{1}{4} \][/tex]
So, option D can be solved using the inverse property of logarithms.
Among the given options (A, B, C, and D), all can be solved using the inverse property. However, since we have determined that the correct answer is B ([tex]\(e^{3.5} = e^{2x}\)[/tex]), [tex]\(e^{3.5} = e^{2x}\)[/tex] can indeed be solved by using the inverse property of the natural logarithm.
Thus, the correct answer is:
[tex]\[ \boxed{B} \][/tex]
Let's examine each option one by one:
A. [tex]\(\log_2 x = \log_2 6\)[/tex]
Using the inverse property of logarithms, we can exponentiate both sides with base 2:
[tex]\[ 2^{\log_2 x} = 2^{\log_2 6} \][/tex]
This simplifies to:
[tex]\[ x = 6 \][/tex]
So, option A can be solved using the inverse property of logarithms.
B. [tex]\(e^{3.5} = e^{2x}\)[/tex]
Using the inverse property of the exponential function [tex]\(e\)[/tex], we take the natural logarithm of both sides:
[tex]\[ \ln(e^{3.5}) = \ln(e^{2x}) \][/tex]
This simplifies to:
[tex]\[ 3.5 = 2x \][/tex]
By solving for [tex]\(x\)[/tex], we get:
[tex]\[ x = \frac{3.5}{2} = 1.75 \][/tex]
So, option B can also be solved using the inverse property of the exponential function.
C. [tex]\(10^{2x} = 10^{-6}\)[/tex]
Using the inverse property of the exponential function with a base of 10, we can take the common logarithm (base 10) of both sides:
[tex]\[ \log_{10}(10^{2x}) = \log_{10}(10^{-6}) \][/tex]
This simplifies to:
[tex]\[ 2x = -6 \][/tex]
By solving for [tex]\(x\)[/tex], we get:
[tex]\[ x = -3 \][/tex]
So, option C can be solved using the inverse property of the exponential function with base 10.
D. [tex]\(\log_4 x = -1\)[/tex]
Using the inverse property of logarithms, we can exponentiate both sides with base 4:
[tex]\[ 4^{\log_4 x} = 4^{-1} \][/tex]
This simplifies to:
[tex]\[ x = \frac{1}{4} \][/tex]
So, option D can be solved using the inverse property of logarithms.
Among the given options (A, B, C, and D), all can be solved using the inverse property. However, since we have determined that the correct answer is B ([tex]\(e^{3.5} = e^{2x}\)[/tex]), [tex]\(e^{3.5} = e^{2x}\)[/tex] can indeed be solved by using the inverse property of the natural logarithm.
Thus, the correct answer is:
[tex]\[ \boxed{B} \][/tex]
