Question

Select the correct answer. Which exponential equation is equivalent to this logarithmic equation? \( \log x = 4 \)?

a. \( x = 10^4 \)
b. \( x = 4^4 \)
c. \( x = e^4 \)
d. \( x = \frac{1}{4} \)

Answer 1

the correct answer is: a. \( x = 10^4 \)

Step-by-step explanation

To convert a logarithmic equation to an exponential equation, we use the definition of the logarithm. The definition states that if \( \log_b(a) = c \), then the equivalent exponential equation is \( b^c = a \).

Looking at our logarithmic equation, \( \log x = 4 \), we can see there is no base written next to the log. When this happens, it is understood that the base is 10, because, by convention, the "log" function without a specified base implies base 10. This is known as the common logarithm.

Let's apply the definition to our equation:
Given \( \log x = 4 \), this is equivalent to \( 10^4 = x \).

Let's look at our options to see which one matches:

a. \( x = 10^4 \) — This is exactly the exponential form of our given logarithmic equation.
b. \( x = 4^4 \) — This implies that the base of the logarithm is 4, which does not match our equation.
c. \( x = e^4 \) — This implies a natural logarithm (base e), which also does not match our equation.
d. \( x = \frac{1}{4} \) — This does not represent an exponential form and is not equivalent to the logarithmic statement provided.

Therefore, the correct answer is:

a. \( x = 10^4 \)