Introduction to Logarithms
The importance of having inverse functions leads us to the introduction to logarithms as the inverses of exponential functions. Solving exponential equations often involves using the ideas presented in the introduction to logarithms to simplify or access the variable for manipulation. To take the log of an exponential function helps us to use the exponentiated term.
Function Notation with Logs and Exponentials
Function notation is used frequently in science to express functions that contain logs and exponents. We learn to use function notation with logs and exponentials in order to solve problems such as computing compounding interest. We can solve these problems written in function notation with logs and exponentials using techniques from solving exponential and log equations.
Logarithmic Product
There are a few rules that can be used when solving logarithmic equations. One of these rules is the logarithmic product rule, which can be used to separate complex logs into multiple terms. Other rules that can be useful are the quotient rule and the power rule of logarithms. The logarithmic product rule is important and is used often in calculus when manipulating logs and simplifying terms for derivation.
Logarithm Power Rule
When evaluating logarithmic equations, the logarithm power rule can be a useful tool. The logarithmic power rule can also be used to access exponential terms. When a logarithmic term has an exponent, the logarithm power rule says that we can transfer the exponent to the front of the logarithm. Along with the product rule and the quotient rule, the logarithm power rule can be used for expanding and condensing logarithms.
Condensing Logarithms
When evaluating logarithmic equations, we can use methods for condensing logarithms in order to rewrite multiple logarithmic terms into one. Condensing logarithms can be a useful tool for the simplification of logarithmic terms. When condensing logarithms we use the rules of logarithms, including the product rule, the quotient rule and the power rule.
Change of Base Formula
When we encounter logarithms with bases not of the common or natural logarithm, we often need the change of base formula. The change of base formula allows us to convert a logarithm from one base to another. By using the change of base formula, we can change a logarithmic term to allow us to input it into a calculator. Most calculators only accept logarithms of base 10 or base e.
Solving Logarithmic Equations
Just as we can use logarithms to access exponents in exponential equations, we can use exponentiation to access the insides of a logarithm. Solving logarithmic equations often involves exponentiating logarithms in order to get rid of the log and access its insides. Sometimes we can use the product rule, the quotient rule, or the power rule of logarithms to help us with solving logarithmic equations.
Compound Interest
One real world application of exponential equations is in compound interest. The formula for compound interest with a finite number of calculations is an exponential equation. We can solve for a parameter of this equation, and can use logarithms to access parameters in the exponent. Students may be asked to solve compound interest problems with interest compounded biannually, monthly, or daily.
Exponential Decay
Exponential decay refers to an amount of substance decreasing exponentially. Exponential decay is a type of exponential function where instead of having a variable in the base of the function, it is in the exponent. Exponential decay and exponential growth are used in carbon dating and other real-life applications.
Simple Logarithmic Equations
In order to understand solving logarithmic equations, students must understand the basics of logarithms, and how to use exponentiation to access the terms inside the logarithm. Some more complicated instances of solving simple logarithmic equations require knowledge of the product, quotient and power rules of logarithms in order to simplify complex terms.
Graphing Logarithmic Functions
In science classes we will often find ourselves graphing logarithmic functions to describe situations such as motion or speed over time. When trying to identify these situations as those seen in graphing logarithmic functions, it is important to be able to recognize these graphs. It is also important to recognize graphs of exponential functions and their importance as the logarithmic inverse.
Quotient Rule of Logarithms
When evaluating logarithms the logarithmic rules, such as the quotient rule of logarithms, can be useful for rewriting logarithmic terms. The quotient rule of logarithms allows us to separate parts of a quotient within a log. The quotient rule of logarithms is useful for expanding and condensing logarithms, along with the product rule and the power rule of logarithms.
Expanding Logarithms
Sometimes we use methods for expanding logarithms in complicated logarithmic terms. Expanding logarithms can be useful for obtaining more simplified terms. When expanding logarithms we use the rules of logarithms, including the power rule, the product rule and the quotient rule. Sometimes we use methods for expanding logarithms when evaluating logarithmic equations.
Common Logarithm
We can use many bases for a logarithm, but the bases most typically used are the bases of the common logarithm and the natural logarithm. The common logarithm has base 10, and is represented on the calculator as log(x). The natural logarithm has base e, a famous irrational number, and is represented on the calculator by ln(x). The natural and common logarithm can be found throughout Algebra and Calculus.
Exponential Equations with Different Bases
Sometimes we are given exponential equations with different bases on the terms. In order to solve these equations we must know logarithms and how to use them with exponentiation. We can access variables within an exponent in exponential equations with different bases by using logarithms and the power rule of logarithms to get rid of the base and have just the exponent.
Solving a Logarithmic Equation with Multiple Logs
When given a problem on solving a logarithmic equation with multiple logs, students should understand how to condense logarithms. By condensing the logarithms, we can create an equation with only one log, and can use methods of exponentiation for solving a logarithmic equation with multiple logs. This requires knowledge of the product, quotient and power rules of logarithms.
Continuous Compound Interest
Problems that involve continuous compound interest use a different equation from problems that have finitely compounded interest, but the continuous compound interest equation is also an exponential equation. We use many of the same methods for calculating continuous compound interest as we do finitely compounded interest. To calculuate compound interest, we can use logarithms and methods for solving exponential equations.
Evaluating Logarithms
In order to do problems on evaluating logarithms in terms of known quantities, we need to understand the rules of logarithms such as the product, the quotient and the power rules. When we say evaluating logarithms in terms of known quantities, we mean that we want to be able to rewrite a complex logarithm so that we can recognize certain simpler logarithmic terms within it.
