Logarithms are powerful tools for simplifying complex calculations and solving equations. They follow specific rules that allow us to manipulate expressions, making them easier to work with. These properties are essential for tackling a wide range of mathematical problems.
Logarithmic functions have unique characteristics that set them apart from other functions. Understanding their graphs, domains, and ranges helps us apply them to real-world situations, from measuring earthquake intensity to modeling population growth.
Properties of Logarithms
Product rule for logarithms
States logb(M⋅N)=logb(M)+logb(N)
Splits the logarithm of a product into the sum of logarithms
Useful for simplifying complex logarithmic expressions (log2(x⋅y)=log2(x)+log2(y))
Quotient rule for logarithms
States logb(NM)=logb(M)−logb(N)
Rewrites the logarithm of a quotient as the difference of logarithms
States logb(x)=loga(b)loga(x), where a is any base
Calculates logarithms with different bases using a common base (usually 10 or e)
Useful when a calculator doesn't have a button for the desired base (log5(x)=log10(5)log10(x))
Logarithmic vs exponential forms
Logarithmic and exponential forms are inverses of each other (inverse functions)
If logb(x)=y, then by=x
Converting between forms helps solve equations (log2(8)=3⇔23=8)
Solving logarithmic equations
Uses the properties of logarithms and the relationship between logarithmic and exponential forms
Steps:
Isolate the logarithm on one side of the equation
Convert the logarithmic equation to its exponential form
Solve for the variable
Example: Solve log3(x+1)=2
log3(x+1)=2
32=x+1
9=x+1⇒x=8
Graphing of logarithmic functions
General form: f(x)=logb(x), where b>0 and b=1
Domain: (0,∞), range: (−∞,∞)
Vertical asymptote at x=0, no horizontal asymptote
y-intercept at (1,0)
Increasing if b>1, decreasing if 0<b<1
Applications of logarithmic properties
Used in various fields (chemistry, physics, biology)
scale (chemistry): measures acidity or basicity of a solution
Decibel scale (physics): quantifies sound intensity or power levels
Population growth (biology): models exponential growth or decay
Richter scale (geology): measures the magnitude of earthquakes
An increase of 1 on the Richter scale corresponds to a tenfold increase in the amplitude of seismic waves
Key Components of Logarithmic Functions
Base: The number b in logb(x), which determines the function's behavior
Domain: All positive real numbers, excluding zero (0,∞)
Range: All real numbers (−∞,∞)
Asymptote: The vertical line x=0, which the function approaches but never crosses
Key Terms to Review (6)
Change-of-base formula: The change-of-base formula allows you to evaluate logarithms with any base using logarithms of different bases. It is given by the formula $\log_b(a) = \frac{\log_c(a)}{\log_c(b)}$, where $c$ is any positive number.
One-to-one: A one-to-one function is a function where each input corresponds to exactly one unique output, and each output corresponds to exactly one unique input. This ensures that no two different inputs produce the same output.
PH: pH is a logarithmic scale used to specify the acidity or basicity of an aqueous solution. It is defined as the negative logarithm of the hydrogen ion concentration.
Power rule for logarithms: The power rule for logarithms states that $\log_b(x^r) = r \cdot \log_b(x)$, where $b$ is the base of the logarithm. This property allows you to move the exponent in a logarithmic expression to the front as a multiplier.
Product rule for logarithms: The product rule for logarithms states that the logarithm of a product is equal to the sum of the logarithms of the factors. Mathematically, $\log_b(xy) = \log_b(x) + \log_b(y)$.
Quotient rule for logarithms: The quotient rule for logarithms states that the logarithm of a quotient is equal to the difference of the logarithms of the numerator and denominator: $\log_b\left(\frac{M}{N}\right) = \log_b(M) - \log_b(N)$, where $b$ is the base.