Problem 10 - College Algebra

Question

Write each equation in its equivalent logarithmic form. $$5^{4}=625$$

Step-by-Step Solution

Answer

\(\log_{5}625 = 4\)

Step 1: STEP 1: Understand the Concept

Exponential and logarithmic functions are inverses of each other. Here's how we can write an exponential function as a logarithmic function and vice-versa: if \(a = b^{c}\), then the equivalent logarithmic form is \(log_b a = c\). Essentially central numbers stay the same, while outer numbers switch positions.

Step 2: STEP 2: Write equation in exponential form

The given equation \(5^{4}=625\) is already in exponential form where \(b = 5\), \(c = 4\) and \(a = 625\). Next, employ the relationship explained in Step 1 to convert this to logarithm form.

Step 3: STEP 3: Write equation in log form

Applying the definition of logarithm, the equation transforms as \(\log_{5}625 = 4\). In other words, it can be read as: 'log to base 5 of 625 is equal to 4', which essentially means that the number 5 must be raised to the power of 4 to get 625.