9AB - Laws of Logarithms & Graphing Logarithmic Functions Video lesson
Evaluating Logs Without a Calculator
In this lesson, we begin with evaluating logarithms, particularly those that do not have base 10 or are not easily calculated using a calculator. The aim is to transition logarithmic expressions into exponential form. For example, evaluating the logarithm of 36 to the base 6 starts by setting the equation as:
[ \log_{6}(36) = x ]
This can be rewritten in exponential form:
[ 6^{x} = 36 ]
To solve for x, we find a common base for both sides. Expressing 36 as (6^2), we can equate the exponents:
[ x = 2 ]
Thus, ( \log_{6}(36) = 2 ).
Example B: Evaluating Log of Fractional Bases
Next, we evaluate ( \log_{8}(1/4) ). Setting this to x:
[ 8^{x} = \frac{1}{4} ]
We can express ( \frac{1}{4} ) as (4^{-1} ), giving:
[ 8^{x} = 4^{-1} ]
Next, convert both sides to a common base:
[ (2^{3})^{x} = (2^{2})^{-1} ]
This simplifies to:
[ 2^{3x} = 2^{-2} ]
By equating the exponents, we find:
[ 3x = -2 \quad \Rightarrow \quad x = -\frac{2}{3} ]
Thus, ( \log_{8}(1/4) = -\frac{2}{3} ).
Example C: Basic Logarithmic Identity
Then we consider ( \log_{8}(8^{5}) ). Rewriting, we have:
[ 8^{x} = 8^{5} ]
Since the bases are the same, x equals 5. This showcases that ( \log_{a}(a^{b}) = b ), where a is the base.
Logarithms of Negative and Zero Values
Evaluating ( \log_{3}(-9) ) reveals that there is no base that defines a logarithm for negative numbers:
[ 3^{x} = -9 \Rightarrow \text{ not possible} ]
Hence, logarithms of negative numbers do not exist. Additionally, ( \log_{10}(0) ) also does not yield a valid result since no exponent applied to 10 can produce 0:
[ 10^{x} = 0 \Rightarrow \text{ does not exist} ]
Evaluating Logarithms at One
For ( \log_{ rac{2}{3}}(1) ), we rearrange:
[ \frac{2}{3}^{x} = 1 ]
Recognizing that any base raised to the power of 0 equals 1, we arrive at:
[ x = 0 ]
Graphing Logarithmic Functions
To graph a logarithmic function effectively, we utilize its inverse, the exponential function. For example, for( y = \log_{2}{x} ), we graph its inverse ( y = 2^{x} ).
Creating a Table of Values
For the exponential function, we select various x values to compute y values:
( x = -1 \quad \Rightarrow \quad y = \frac{1}{2} )
( x = 0 \quad \Rightarrow \quad y = 1 )
( x = 1 \quad \Rightarrow \quad y = 2 )
( x = 2 \quad \Rightarrow \quad y = 4 )
Using these to reverse the table for the logarithmic function:
( x = \frac{1}{2} \quad y = -1 )
( x = 1 \quad y = 0 )
( x = 2 \quad y = 1 )
( x = 4 \quad y = 2 )
Thus, a log function graph can be approximated accordingly.
Analyzing Properties of Logarithmic Functions
Domain and Range
For the function ( y = \log_{2}(x) ):
Domain: All x values such that ( x > 0 )
Range: All y values from negative infinity to positive infinity.
Asymptotes and Intercepts
Vertical Asymptote: At x = 0 (the y-axis)
X-Intercept: At (1, 0)
The Laws of Logarithms
Moving on to the laws of logarithms, which are crucial for simplifying logarithmic expressions.
1. Logarithm of a Product (Product Law)
The first law states that: [ \log_{a}(mn) = \log_{a}(m) + \log_{a}(n) ] This law applies when multiplying two factors within the log.
Example
For ( \log_{10}(2) + \log_{10}(50) ): Using the product law leads to: [ \log_{10}(2 \times 50) = \log_{10}(100) = 2 ]
2. Logarithm of a Quotient (Quotient Law)
The second law applies when dealing with division: [ \log_{a}(\frac{m}{n}) = \log_{a}(m) - \log_{a}(n) ]
3. Logarithm of a Power (Power Law)
If a number is multiplied by the log: [ n \cdot \log_{a}(m) = \log_{a}(m^{n}) ]
Combined Example
For an expression ( \log_{3}(x) + 3 \cdot \log_{3}(y) - 4 \cdot \log_{3}(z) ):
Transform the logs as needed: [ \log_{3}(x) + \log_{3}(y^{3}) - \log_{3}(z^{4}) ]
Combine them using the product and quotient laws: [ \log_{3}(\frac{xy^{3}}{z^{4}}) ]
Important Notes on Logarithmic Expressions
When expanding logs, all logs should preferably start positive to avoid complications in combining them. If combining, always ensure to factor appropriately to maintain the correct sign throughout extrapolation. Studying and practicing these laws will enhance your ability to tackle logarithmic functions in various contexts.