Using Properties of Logs to Solve Logarithmic Equations
Description/Explanation/Highlights
Video Description
This video explains how properties of logarithms can be used to solve logarithmic equations.
Steps and Key Points to Remember
To use properties of logarithms to solve equations, follow these steps.
- Properties of logarithms are applied to rewrite logarithmic expressions to make solving easier.
- To solve a problem like \(\log_x(4)+\log_x(16)=2\), we first apply the log of products property to rewrite the left side as a single logarithm.
- This results in multiplying the two arguments inside a single log i.e. \(\log_x(4\cdot 16)=2\).
- Since \(4\cdot 16=64\), we can take the single log of 64 by rewriting the equation in exponential form: \(x^2=64\) and taking the square root of both sides to get \(x=8\).
- We can follow the same process with subtraction by applying the log of quotients property i.e. \(\log_2(x)-\log_2(4)=2\) becomes \(\log_2(\frac{x}{4})=2\).
- Now rewrite in exponential form to get an equation that can be solved: \(2^2=\frac{x}{4}\) and by multiplying both sides of the equation by 4, \(x=16\)
- \(2\log_9(3)=x\) can be solved by first moving the 2 back to the power inside the log i.e. \(\log_9(3)^2=x\) which becomes \(\log_9(9)=x\).
- Using the log property that says, if the argument and base of a log are the same, the log is equal to the power, and since the power is not present and therefore = 1, x must equal 1.
Here are some key points to keep in mind when solving equations by applying properties of logarithms.
- Multiplication inside a single Log becomes addition of two Logs.
- Division inside a single Log becomes subtraction of two Logs.
- When a single Log is raised to a power, the power can be moved to the front of the Log and multiplied and when a number is multiplied by a Log, it can be moved to the power inside of the log..
- If the base and argument of a Log are the same, the Log is equal to the power to which the Log is raised. if there is no power, it is assumed to be 1 and Log = 1.
- Multiple Log properties are also often used to simplify Logs and write multiple Logs into a single Log.
Video Highlights
- 00:00 Introduction
- 00:22 \(\log_x(4)+\log_x(16)=2\) example of using the Log of a product property
- 01:38 \(\log_2(x)-\log_2(4)=2\) example using Log of a quotient property
- 02:42 \(2\log_9(3)=x\) example combining the Log of a power property and finding logs when the base and argument are the same.
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