In effect, you will practice the skills necessary to transform the above Eq2 into Eq3.ĭefinitions Let 𝑏 > 0, D1. The remainder of this less will focus on writing equivalent exponential and logarithmic statements, rather than working between an original statement and its inverse. Inverse of Original Function In Exponential Form: (Eq2) 𝑥 = 10𝑦 (Eq3) Warning: economists usually mean natural log (a log with base 𝑒) and NOT the base-ten log when they write the log without a base. Common logs can be written without showing the 10. Logarithms with base 10 are considered common logs. As shown on the graph, the base number “10” is omitted and the logarithm is written as 𝑦 = log 𝑥. ” The below picture shows 𝑦 = 10𝑥, and its inverse 𝑦 = log10 𝑥. The above exponential statement, 𝑥 = 𝑏 𝑦 is equivalent to the logarithmic statement, 𝑦 = 𝑙𝑜𝑔𝑏 (𝑥), and is pronounced, “y equals log base b of x.” The value of the subscript “b” is the “base of the logarithm,” just as “b” is the base of the exponential in the expression “𝑏 𝑦. Equivalent Exponential and Logarithmic Forms 𝑥 = 𝑏 𝑦 is equivalent to 𝑦 = 𝑙𝑜𝑔𝑏 (𝑥) * For any positive base b, where b≠1. The statements 𝑥 = 𝑏 𝑦 and 𝑦 = 𝑙𝑜𝑔𝑏 (𝑥) are equivalent and both represent the inverse of 𝑦 = 𝑏 𝑥. However written as 𝑥 = 𝑏 𝑦, “x is a function of y.” To express the inverse where “y is a function of x,” write 𝑦 = 𝑙𝑜𝑔𝑏 (𝑥).

The exponential function 𝑦 = 𝑏 𝑥, is a one-to-one function1, so its inverse 𝑥 = 𝑏 𝑦 is a function. Logarithms Logarithms are the inverse of exponentials.

Inverse of log10