Problem 120 - College Algebra

Question

Describe the relationship between an equation in logarithmic form and an equivalent equation in exponential form.

Step-by-Step Solution

Answer

Logarithmic and exponential forms of equations are interchangeable. For an equation in exponential form \(a = b^c\), its equivalent logarithmic form would be \(\log_b a = c\). Conversely, for a logarithmic equation \(\log_b a = c\), its equivalent exponential form would be \(a = b^c\).

Step 1: Defining the Forms

Firstly, understand the equation in its exponential form: \(a = b^c\) where a is the result of b (the base) raised to the power c (the exponent). 'b' is always a positive number other than 1, 'a' is positive and 'c' can be any real number. An equivalent logarithmic form of this equation is written as \(\log_b a = c\) in which \(\log_b a\) denotes the number you would need to raise 'b' to get 'a'.

Step 2: Converting Exponential to Logarithmic form

If we have an exponential equation \(a = b^c\), it could be converted to logarithmic form as follows: Start by identifying 'a', 'b' and 'c' from the exponential form. 'b' is the base of the logarithm, 'a' is the number we're taking the log of, and 'c' is the value the logarithm should equal. Hence we can write this as a logarithmic equation \(\log_b a = c\)

Step 3: Converting Logarithmic to Exponential form

The process of conversion from logarithmic to exponential form is just the reverse of the above process. If we have a logarithmic equation \(\log_b a = c\), 'b' is the base raised to equate 'c' to give 'a'. Thus the equation becomes \(a = b^c\) in exponential form