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Logarithms Explained Rules & Properties, Condense, Expand, Graphing & Solving Equations Introduction

The Organic Chemistry Tutor・2 minutes read

Logarithms provide the relationship between exponents and numbers, allowing for the conversion of exponential equations into logarithmic form and vice versa, as well as applying various properties and formulas for calculating and simplifying expressions. In addition, understanding how to graph exponential functions involves finding key points, horizontal asymptotes, and recognizing the increasing or decreasing rate of growth associated with these functions.

Insights

Logarithms with base 2, 3, 4, 5, 6, 7, and 10 provide essential relationships between numbers and their exponents, helping to simplify complex calculations and understand exponential growth.
Properties of logarithms include combining logs through addition and subtraction, expanding logs by distributing exponents, and utilizing the change of base formula to convert between different bases, enhancing flexibility in mathematical operations.
Solving logarithmic and exponential equations often involves converting between forms, applying properties of logs, and graphing functions to visualize their behavior, offering powerful tools for problem-solving and understanding mathematical relationships.

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Recent questions

What is the definition of a logarithm?
Logarithm is the inverse of exponentiation.
How do you solve logarithmic equations?
Convert to exponential form and solve for the variable.
What is the natural logarithm?
The natural logarithm uses base e ≈ 2.71828.
How do you graph an exponential function?
Identify key points, asymptotes, and plot accordingly.

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Summary

00:00

Understanding Logarithms: A Brief Overview

Log base 2 of 4 is 2, as 2 to the power of 2 equals 4.
Log base 2 of 8 is 3, as 2 to the power of 3 equals 8.
Log base 3 of 9 is 2, as 3 squared equals 9.
Log base 4 of 16 is 2, as 4 squared equals 16.
Log base 3 of 27 is 3, as 3 to the power of 3 equals 27.
Log base 2 of 32 is 5, as 2 to the power of 5 equals 32.
Log base 5 of 125 is 3, as 5 to the power of 3 equals 125.
Log base 6 of 36 is 2, as 6 squared equals 36.
Log base 2 of 64 is 6, as 2 to the power of 6 equals 64.
Log base 3 of 81 is 4, as 3 to the power of 4 equals 81.
Log base 7 of 49 is 2, as 7 squared equals 49.
Log base 3 of 1 is 0, as any number raised to the power of 0 equals 1.
Log of 10 is 1, as 10 to the power of 1 equals 10.
Log of 100 is 2, as 10 squared equals 100.
Log of 1000 is 3, as 10 to the power of 3 equals 1000.
Log of 1 million is 6, as it has 6 zeros.
Log of 0.1 is -1, log of 0.01 is -2, and log of 0.001 is -3.
Log of negative numbers or 0 does not exist.
Log base 4 of 16 is 2, log base 4 of 1 over 16 is -2, log base 16 of 4 is 1/2, and log base 16 of 1 over 4 is -1/2.
Change of base formula: log base a of b = log b / log a.
Properties of logs: log a + log b = log ab, log a - log b = log a / b, log a^2 = 2 log a.
Combining logs: log X + log Y - log Z = log XY / Z, - log X + 3 log Y - 4 log Z = log Y^3 / XZ^4.
Expanding logs: log R^2 S^5 Z^6 = 2 log R + 5 log S - 6 log Z, log cube root x^2 Y / Z^4 = (2/3) log x + log Y - (4/3) log Z.

20:02

Expanding, Simplifying, and Solving Logarithmic Expressions

To expand a logarithmic expression, distribute the 1/3 to each of the three new logs: log x^2 + log y - 4 log z.
Leave C on the bottom as log z^4, then move the exponents to the front for the final answer: (1/3)(log x) + log y - 4 log z.
Practice by converting radical expressions into exponents: log x^2 + log y^(1/2) - log z^(4/3).
Move the exponents back to the front for the final answer: 2 log x + (1/2) log y - (4/3) log z.
The natural log of 1 is always 0, the natural log of e is always 1, and ln e to the fifth power is 5.
For expressions like e^(ln 7) and e^(3 ln x + 2), simplify by canceling out the bases to get 7 and x^3, respectively.
Simplify expressions like 5^(log base 5 of y^8) to y^8 and 3^(log base 3 of 6/2) to 3^1.
To simplify log equations, combine logs by division: log base 3 of 6/2 simplifies to 1.
Convert exponential equations into logarithmic form: 3^3 = 27 becomes log base 3 of 27 = 3.
Solve logarithmic equations by converting to exponential form and solving: log base x of 27 = 3 simplifies to x = 3.

41:30

"Equations Solved: X = 1, 10, 7"

To solve for X in the equation 2x + 1 = x + 8, subtract 19 from both sides, then divide by 2 to find X = -19/2.
Plugging in 1 into the equation results in a valid solution, while using -19/2 leads to an extraneous solution, making the final answer X = 1.
Checking the work by substituting X = 1 back into the equation confirms the correctness of the solution.
Solving the equation log base 2 of (x + 6) - log base 2 of (x - 8) = 3 involves combining the logs into a single expression and cross-multiplying to find X = 10.
When dealing with log equations with the same base, set the insides equal to each other to solve for X, as demonstrated in log base 2 of (x + 5) = log base 2 of (3x - 9), resulting in X = 7.
For Ln X = 5, convert it to exponential form to find X = e^5.
In the equation Ln X + 3 Ln X = 0, simplify to 4 Ln X = 0, leading to X = 1.
To solve Ln (X - 3) = 2, convert to exponential form and add 3 to find X = e^2 + 3.
When dealing with exponential equations, like 8^(3x + 4) = 16^(2x - 3), convert bases to a common one (2) and solve for X to get X = 24/5.
Finding X in 2^X = 7 involves using natural logs to solve for X = ln 7 / ln 2, approximately equal to 2.80735.

01:01:34

Graphing Exponential Functions with Key Points

To graph an exponential function, start with y = 2^(x-3) + 1.
Find key points by setting the exponent to 0 and 1, giving x values of 3 and 4.
Calculate y values for x = 3 and x = 4, resulting in points (3, 2) and (4, 3).
The horizontal asymptote for this function is y = 1.
Plot the horizontal asymptote at y = 1 and points (3, 2) and (4, 3) to graph the function.
Exponential functions increase at an increasing rate.
For another exponential function, set the exponent to 0 and 1, yielding x values of -1 and 0.
Calculate y values for x = -1 and x = 0, resulting in points (-1, 0.7) and (0, -1).
The horizontal asymptote for this function is y = -2.
Plot the horizontal asymptote at y = -2 and points (-1, 0.7) and (0, -1) to graph the function.

01:21:59

Exponential Function Plotting and Symmetry Examples

Plotting the exponential function involves starting from the asymptotes and following the points, ensuring symmetry with the inverse function across the line y equals x, with specific examples of the functions e to the X plus 2 and log X minus 2.

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