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Logarithmic Function
The logarithmic function with base , denoted as , is formally defined as the inverse of the exponential function . It is defined for , where the base and . Although the one-to-one property of an exponential function guarantees that it has an inverse, there is no basic algebraic operation to solve an equation like for . Therefore, the logarithmic function is uniquely introduced to represent this inverse relationship, where the notation mathematically 'undoes' the exponential function.
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Ch.10 Exponential and Logarithmic Functions - Intermediate Algebra @ OpenStax
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Logarithmic Function
A small business owner uses a financial tool to compare two different savings accounts that grow at the same monthly rate. The tool generates the equation to determine when the account balances will be identical. Which property of exponential functions allows the owner to conclude that ?
An operations manager at a logistics company uses an exponential function to model package sorting times. When analyzing the model's equations to determine when two processing times with the same base rate will be equal (), the manager applies the rule that the inputs must also be equal (). Graphically, this one-to-one property is demonstrated because the exponential model passes the ________.
A data analyst is validating an exponential growth model used to forecast company revenue. Match each aspect of the exponential function's "one-to-one" nature with its corresponding mathematical or graphical description.
An environmental scientist is using an exponential model to track the decay of a substance over time. If the model leads to the equation (where the base and ), is it true that the one-to-one property of exponential functions allows the scientist to conclude that ?
Applying the One-to-One Property in Technical Modeling
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Equivalence of Logarithmic and Exponential Equations
Shape of the Graph of a Logarithmic Function where
Point on the Graph of a Logarithmic Function
Point on the Graph of a Logarithmic Function
Domain of a Logarithmic Function
Range of a Logarithmic Function
Vertical Asymptote of the Graph of a Logarithmic Function
Natural Logarithmic Function
Common Logarithmic Function
Quotient Property of Logarithms
Power Property of Logarithms
Example 10.31: Using the Quotient Property of Logarithms
Try It 10.61: Writing Logarithms as a Difference of Logarithms
Try It 10.62: Applying the Quotient Property of Logarithms
Example 10.33: Expanding a Logarithm Using the Product and Power Properties
Try It 10.65: Expanding a Logarithm Using the Product and Power Properties
Try It 10.66: Expanding a Logarithm Using the Product and Power Properties
Product Property of Logarithms
Inverse Properties of Logarithms
Logarithm of 1 Property
Logarithm of the Base Property
Example 10.34: Expanding a Logarithm with a Radical
Try It 10.67: Expanding a Logarithm with a Radical
Try It 10.68: Expanding a Logarithm with a Radical
Condensing Logarithmic Expressions
Mirror Image Relationship between Logarithmic and Exponential Functions
-intercept of the Graph of a Logarithmic Function
One-to-One Property of Logarithmic Equations
Shape of the Graph of a Logarithmic Function where
-intercept of the Graph of a Logarithmic Function
As a data analyst at your company, you are modeling the growth of customer acquisitions using the exponential function . To determine the specific timeframe needed to reach a target number of customers, you must use the logarithmic function. Match the following logarithmic terms used in your analysis with their correct mathematical definitions or requirements.
A marketing specialist is analyzing the growth of a social media campaign. The reach of the campaign is modeled by the exponential function , where is the total reach and is the time in hours. To find the time required to reach a specific audience size, the specialist must use the inverse function. Which of the following is the correct logarithmic expression for ?
In a corporate research setting, a scientist is using the logarithmic function to analyze experimental data. True or False: This function is formally defined as the inverse of the exponential function and is defined for all real values of .
Defining Logarithmic Functions for Financial Modeling
Defining Logarithmic Functions for Operational Modeling