Problem 1 - College Algebra

Question

Write each equation in its equivalent exponential form. $$4=\log _{2} 16$$

Step-by-Step Solution

Answer

The equivalent exponential form of the given logarithmic equation \(4 = \log_2 16\) is \(2^4 = 16\).

Step 1: Identify the base, exponent, and result in the logarithmic equation

In the given logarithmic equation \(\log_2 16 = 4\), the base of the logarithm is 2, the result of the logarithm (the number that the logarithm is equal to) is 4, and the number that the logarithm is acting on (inside the logarithm) is 16.

Step 2: Apply the relationship between logarithms and exponents

A logarithmic equation \(\log_b a = c\) can be converted into the exponential equation \(b^c = a\). Here, \(b\) is the base, \(a\) is the number inside the logarithm, and \(c\) is the result. For the given equation \(\log_2 16 = 4\), this gives \(2^4 = 16\).

Step 3: Check the solution

After transforming the logarithmic equation into an exponential one, it can be checked by ensuring that the equation holds true. For \(2^4 = 16\), raising 2 to the power of 4 indeed gives 16, so the exponential form is correct.