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Logarithms Review - Exponential Form - Graphing Functions & Solving Equations - Algebra

The Organic Chemistry Tutor・55 minutes read

Logarithms cover evaluating, properties, expanding, condensing, change of base formula, conversions, equations, word problems, compounded interest, and graphing. Calculations revolve around recognizing powers and bases, utilizing the change of base formula, natural logs, and graphing functions to determine growth or decay.

Insights

Evaluating logarithms involves determining the power needed to reach a number by multiplying a base, such as log base 2 of 8 being 3 because 2^3 equals 8, showcasing the fundamental concept of logarithms.
The change of base formula provides flexibility in calculations by allowing the conversion of logarithms between different bases, exemplified through the formula log of b divided by log of a equals the log base a of b, demonstrating a key method to simplify logarithmic operations.

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Recent questions

What are logarithms?
Logarithms are mathematical functions that represent the inverse operations of exponentiation. They help in solving equations involving exponential functions and are used to convert between exponential and logarithmic forms.
How do you evaluate logarithms?
Evaluating logarithms involves determining the power to which a base must be raised to reach a given number. For example, log base 2 of 8 is 3 because 2 raised to the power of 3 equals 8.
What is the change of base formula?
The change of base formula allows converting logarithms between different bases. It states that log base b of x can be expressed as log x divided by log b, providing flexibility in calculations involving logarithmic functions.
How do you solve log equations?
To solve log equations, convert them to exponential form and solve for the variable. In cases where bases differ, taking the natural log of both sides can help simplify the equation and find the solution.
How do you graph exponential functions?
Graphing exponential functions involves identifying asymptotes, choosing points, and plotting them accordingly. Understanding the behavior of exponential functions helps in determining growth or decay trends based on the graph.

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Summary

00:00

Understanding Logarithms: Evaluating, Converting, and Graphing

Logarithms are discussed in the video, covering topics such as evaluating logs, properties of logs, expanding and condensing logs, the change of base formula, converting between exponential and log form, solving equations, word problems, compounded interest, and graphing log and exponential functions.
Evaluating logs involves determining the power needed to reach a number by multiplying a base, such as log base 2 of 8 being 3 because 2^3 equals 8.
Log base 3 of 27 is found by recognizing that 3^3 equals 27, requiring 3 threes to reach 27.
Log base 2 of 32 is calculated by understanding that 2^5 equals 32, necessitating 5 twos to reach 32.
Log base 5 of 25 is determined by realizing that 5^2 equals 25, needing 2 fives to reach 25.
Logarithms with fractions involve negative exponents, like log base 2 of 1 over 16 being -4 due to 2^-4 equaling 1 over 16.
Logarithms with reversed orders result in fractions, such as log 16 of 2 being 1 over 4 because 16^(1/4) equals 2.
Negative fractions are obtained when the base is larger than the number, like log base 16 of 1 over 2 being -1/4.
Logarithms without a specified base default to base 10, with log of 1 being 0, log of 10 to the fifth being 5, and log of 1000 being 3.
The change of base formula allows converting logs between different bases, with log of b divided by log of a equaling the log base a of b, providing flexibility in calculations.

20:46

Logarithm Conversion and Exponential Equations Simplified

To find log base 2 of 8, use log 8 divided by log 2.
For log base 5 of 100, use log 100 divided by log 5 to get approximately 2.861.
To convert a log to a natural log, use ln x squared y cubed over ln 2.
Change log base 3 of 8 to log 8 over log 3 or ln 8 over ln 3.
Log and natural log are similar, with ln specifically having a base of e.
To convert to exponential form, use the formula a raised to the c power equals b.
For log base 3 of 81, log 81 over log 3 equals 4, as 3 to the fourth power is 81.
Log base 5 of 125 is 3, as 5 to the third power is 125.
Solving log equations involves converting to exponential form and solving for x.
In cases where bases differ, taking the natural log of both sides can help solve exponential equations.

40:24

Solving Exponential Equations with Natural Logarithms

Distribute 2x plus 4 by ln3, resulting in 2x ln3 plus 4 ln3 equals x ln7 minus 3 ln7.
Rearrange the equation to isolate x by moving terms with x to one side and those without x to the other side.
Factor out x to simplify the equation to x times 2 ln3 minus ln7 equals negative 4 ln3 minus 3 ln7.
Solve for x by dividing the terms, yielding x equals negative 4 ln3 minus 3 ln7 divided by 2 ln3 minus ln7.
For base e problems, take the natural log of both sides to simplify the equation.
Simplify the equation by moving terms and solving for x, resulting in x equals ln5.
Use substitution to solve equations involving e to the x, replacing e to the x with y and solving for x.
Take the natural log of both sides to find x, ensuring no negative values in the logarithm.
Solve equations involving division by e to the negative x by cross-multiplying and simplifying.
Graph exponential and logarithmic equations by identifying asymptotes, choosing points, and plotting them accordingly.

59:09

Logarithmic and Exponential Equations: Key Formulas

The vertical asymptote is x = -4, with points at (-3, -2) and (3, -1).
Plot the vertical asymptote at x = -4 and the points (-3, -2) and (3, -1).
For x = -3, log base 3 of 1 is 0, resulting in -2; for x = -1, log base 3 of 3 is 1, resulting in -1.
The range is all real numbers from negative infinity to infinity.
The domain ranges from -4 to infinity based on the vertical asymptote.
For word problems involving logs and exponential equations, use specific equations based on compounding frequency.
For compounding monthly, use the formula with p = 5000, r = 0.08, n = 12, and t = 7 years.
To find the initial investment for weekly compounding, use the formula with p = 10000, r = 0.075, n = 52, and t = 8 years.
For continuous compounding at 8% interest, after 4 years, the account will have $685.64.
To determine how long it takes for an account to double at 9% interest, use the formula and find it to be approximately 7.7 years.

01:18:52

Growth and decay in mathematical expressions

Identifying growth or decay in mathematical expressions: Positive values on the bottom indicate decay, negative exponents signify decay, while positive exponents indicate growth. Graphing the function can help determine if it's increasing or decreasing, with upward trends indicating growth and downward trends indicating decay.

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