Explain the relationship between exponential and logarithmic functions. Describe how to calculate a logarithm to a different base. Identify the hyperbolic functions, their graphs, and basic identities. In this section we examine exponential and logarithmic functions.

Exponents and Logarithms work well together because they "undo" each other (so long as the base "a" is the same): They are "Inverse Functions" Doing one, then the other, gets us back to where we started: Doing ax then loga gives us back x: loga(ax) = x. Doing loga then ax gives us back x: aloga(x) = x.

Sep 17, 2024 · Logarithmic functions are the inverses of exponential functions. The inverse of the exponential function y = ax is x = ay. The logarithmic function y = logax is defined to be equivalent to the exponential equation x = ay. y = logax only under the following conditions: x …

Exponential and logarithmic functions are inverse operations of each other. Exponential functions have a base raised to a variable exponent, while logarithmic functions have a base raised to a constant exponent equal to the variable.

13.4: Logarithmic Functions This section introduces logarithmic functions as the inverses of exponential functions. It covers their properties, common and natural logarithms, and how to evaluate and rewrite logarithmic expressions. The section also explains the relationship between logarithmic and exponential equations, including conversion ...

What are Exponential and Logarithmic Functions? Exponential Function Definition: An exponential function is a Mathematical function in the form y = f (x) = b x, where “x” is a variable and “b” is a constant which is called the base of the function such that b > 1.

1.5.1: The Relationship Between Logarithmic and Exponential Functions. We saw earlier that an exponential function is any function of the form \(f(x)=b^x\), where \(b>0\) and \(b\neq1\). A logarithmic function is any function of the form \(g(x)=\log_b{(x)}\), where \(b>0\) and \(b\neq1\).

Since "the logarithm is the exponent,'' it's no surprise that this translates directly into a fact about the logarithm function. Here are three facts from the example: $\log_a(a^5)=5$, $\ds \log_a(a^3)=3$, $\ds \log_a(a^8)=8$. So $\log_a(a^5a^3)=\log_a(a^8)=8 = 5+3=\log_a(a^5)+\log_a(a^3)$. Now let's make this a bit more general.

Definitions: Exponential and Logarithmic Functions. Exponential functions have the form: where b is the base and x is the exponent (or power). If b is greater than \displaystyle {1} 1, the function continuously increases in value as x increases.

In this section we examine exponential and logarithmic functions. We use the properties of these functions to solve equations involving exponential or logarithmic terms, and we study the meaning and importance of the number e.