Expanding Logarithms Expressions

What is Expanding logarithms?

Expanding logarithms refers to the process of taking a logarithmic expression that is compact or condensed and rewriting it as a sum, difference, or multiple of simpler logarithmic terms. This expansion is based on the properties of logarithms and is useful for simplifying the expression and making it easier to work with, especially when solving logarithmic equations or differentiating and integrating logarithmic functions.

Properties of Logarithms

The key properties of logarithms used for expansion include:

1. Product Property:

The logarithm of a product is the sum of the logarithms of the individual factors.

$ \log_b(xy) = \log_b(x) + \log_b(y) $

2. Quotient Property:

The logarithm of a quotient is the difference of the logarithms of the numerator and the denominator.

$ \log_b\left(\frac{x}{y}\right) = \log_b(x) - \log_b(y) $

3. Power Property:

The logarithm of a power is the exponent times the logarithm of the base.

$ \log_b(x^n) = n \log_b(x) $

This property is particularly useful when dealing with roots, as roots can be expressed as fractional exponents.

4. Change of Base Formula:

The logarithm of a number with one base can be converted into an expression involving logarithms of another base.

$ \log_b(x) = \frac{\log_c(x)}{\log_c(b)} $

This formula is particularly useful when you need to calculate logarithms with a base that is not readily available on a calculator.

5. Logarithm of One:

The logarithm of 1 to any base is zero.

$ \log_b(1) = 0 $

This is true because any non-zero number raised to the power of zero is 1.

6. Logarithm of the Base:

The logarithm of the base itself is always one.

$ \log_b(b) = 1 $

This is because the base raised to the power of one is itself.

7. Natural Logarithm:

$\ln(e^b) = b$

Because $e$ raised to the power of $1$ is $e$, fitting the natural logarithm's definition.

How to Expand logarithms？

Let's expand and simplify the following examples using properties of logarithms:

Example 1: $\log(a^2b^3/c)$

Step 1: Apply Quotient Property $ \log_b\left(\frac{x}{y}\right) = \log_b(x) - \log_b(y) $

$ \log(a^2b^3/c)=\log(a^2b^3) - \log(c) $

Step 2: Apply Product Property $ \log_b(xy) = \log_b(x) + \log_b(y) $

$ \log(a^2b^3) - \log(c)=\log(a^2) + \log(b^3) - \log(c) $

Step 3: Apply Power Property $ \log_b(x^n) = n \log_b(x) $

$ \log(a^2) + \log(b^3) - \log(c)=2\log(a) + 3\log(b) - \log(c) $

Example 2: $\ln\left(\frac{e^3 \cdot x^5}{y^2}\right)$

Step 1: Apply Quotient Property $ \log_b\left(\frac{x}{y}\right) = \log_b(x) - \log_b(y) $

$ \ln\left(\frac{e^3 \cdot x^5}{y^2}\right)=\ln(e^3x^5) - \ln(y^2) $

Step 2: Apply Product Property $ \log_b(xy) = \log_b(x) + \log_b(y) $

$ \ln(e^3x^5) - \ln(y^2)=\ln(e^3) + \ln(x^5) - \ln(y^2) $

Step 3: Apply Power Property $ \log_b(x^n) = n \log_b(x) $**

$ \ln(e^3) + \ln(x^5) - \ln(y^2)=3\ln(e) + 5\ln(x) - 2\ln(y) $

Step 4: **Natural Logarithm: **$\ln(e^b) = b$

$ 3\ln(e) + 5\ln(x) - 2\ln(y)=3 + 5\ln(x) - 2\ln(y) $

Example 3: $\log_5\left(\frac{125x^3}{z}\right)$

Step 1: Apply Quotient Property $ \log_b\left(\frac{x}{y}\right) = \log_b(x) - \log_b(y) $**

$ \log_5(125x^3) - \log_5(z) $

Step 2: Apply Product Property $ \log_b(xy) = \log_b(x) + \log_b(y) $**

$ \log_5(125) + \log_5(x^3) - \log_5(z) $

Step 3: Apply Power Property $ \log_b(x^n) = n \log_b(x) $**

$ \log_5(5^3) + 3\log_5(x) - \log_5(z) $

Step 4: Apply Simplify and Power Property $ \log_b(b) = 1 $ $ \log_b(x^n) = n \log_b(x) $

$\log_5(5^3)=3\log_5(5)=3$

$ \log_5(5^3) + 3\log_5(x) - \log_5(z)=3 + 3\log_5(x) - \log_5(z) $

Example 4: $\log_{10}\left(\frac{100x^4y}{z^2}\right)$

Step 1: Apply Quotient Property

$ \log_b\left(\frac{x}{y}\right) = \log_b(x) - \log_b(y) $**

$ \log_{10}\left(\frac{100x^4y}{z^2}\right)=\log_{10}(100x^4y) - \log_{10}(z^2) $

Step 2: Apply Product Property

$ \log_b(xy) = \log_b(x) + \log_b(y) $

$ \log_{10}(100x^4y) - \log_{10}(z^2)$

$=\log_{10}(100) + \log_{10}(x^4) + \log_{10}(y) - \log_{10}(z^2) $

Step 3: Apply Power Property $ \log_b(x^n) = n \log_b(x) $**

$ \log_{10}(100) + \log_{10}(x^4) + \log_{10}(y) - \log_{10}(z^2)$

$=\log_{10}(10^2) + 4\log_{10}(x) + \log_{10}(y) - 2\log_{10}(z) $

Step 4: Apply Simplify and Power Property

$ \log_b(b) = 1 $

$ \log_b(x^n) = n \log_b(x) $

$\log_{10}(10^2) + 4\log_{10}(x) + \log_{10}(y) - 2\log_{10}(z)$

$ 2 + 4\log_{10}(x) + \log_{10}(y) - 2\log_{10}(z) $

Example 5: $\ln\left(\frac{x^2e^2}{y^3}\right)$

Step 1: Apply Quotient Property

$ \log_b\left(\frac{x}{y}\right) = \log_b(x) - \log_b(y) $**

$ \ln\left(\frac{x^2e^2}{y^3}\right)=\ln(x^2e^2) - \ln(y^3) $

Step 2: Apply Product Property

$ \log_b(xy) = \log_b(x) + \log_b(y) $**

$ \ln(x^2e^2) - \ln(y^3)$

$=\ln(x^2) + \ln(e^2) - \ln(y^3) $

Step 3: Apply Product Property

$ \log_b(xy) = \log_b(x) + \log_b(y) $**

$ \ln(x^2) + \ln(e^2) - \ln(y^3)$

$=2\ln(x) + 2\ln(e) - 3\ln(y) $

Step 4: Natural Logarithm:

$\ln(e^b) = b$

$ 2\ln(x) + 2\ln(e) - 3\ln(y)$

$=2\ln(x) + 2 - 3\ln(y) $