Exponential and logarithmic equations are powerful tools for solving real-world problems. They're used to model growth, decay, and complex relationships in fields like finance, science, and engineering. Understanding these equations is crucial for tackling advanced math and practical applications.
Solving these equations involves techniques like using like bases, applying logarithms, and leveraging their inverse relationship. Mastering these skills opens doors to analyzing compound interest, population growth, radioactive decay, and more. It's a key step in your mathematical journey.
Exponential Equations
Solving exponential equations with like bases
Set exponents equal when exponential equations have the same base on both sides (2x=25, solve by setting x=5)
Rewrite equation using desired base when one side has it and the other does not (3x=9, rewrite 9 as 32 to get 3x=32, then set x=2)
Rewrite both sides using desired base when neither side has it (16=4x, rewrite 16 as 42 to get 42=4x, then set 2=x)
Logarithms for exponential equations
Use logarithms to solve exponential equations with different bases (2x=3, take log2 of both sides to get x=log2(3))
Logarithm base is the same as the exponential base of the side without the variable
Solve equations with base e using natural logarithms (ln) (ex=10, take ln of both sides to get x=ln(10))
Logarithmic Equations
Definition of logarithms in equations
logb(x)=y is equivalent to by=x
Rewrite logarithmic equations in exponential form to solve (log3(x)=4, rewrite as 34=x, so x=81)
Argument of a logarithm (value inside parentheses) must be positive (log2(x−3)=5, rewrite as 25=x−3, so x=35)
One-to-one property for logarithmic equations
If logb(x)=logb(y), then x=y
Equate arguments of logarithms with the same base (log2(x)=log2(3x−1), set arguments equal to get x=3x−1, solve for x=21)
Check for extraneous solutions that make logarithm argument non-positive (log(x)=log(x−2), set x=x−2, solve for x=2, but this makes log(0), so there is no solution)
Properties of Exponential and Logarithmic Functions
Domain and Range
Exponential functions: Domain is all real numbers, range is all positive real numbers
Logarithmic functions: Domain is all positive real numbers, range is all real numbers
Inverse Functions
Exponential and logarithmic functions are inverses of each other
y=bx and y=logb(x) are inverse functions
Asymptotes
Exponential functions have a horizontal asymptote (y = 0 for y=bx where 0<b<1, y = 0 for y=b−x where b>1)
Logarithmic functions have a vertical asymptote (x = 0)
Applications
Real-world applications of exponential equations
Exponential growth and decay models: A(t)=A0ekt (A0 is initial amount, k is growth or decay rate, t is time)
Population starting at 100 grows by 5% annually, find population after 10 years: A(t)=100e0.05t, plug in t=10 to get A(10)≈164.87
Logarithmic scales (Richter scale for earthquake magnitudes, pH scale for acidity)
Earthquake of magnitude 6 is how many times more intense than magnitude 4? Richter scale is logarithmic base 10, difference of 2 corresponds to factor of 102=100
and doubling time problems involving exponential decay or growth
Substance has half-life of 20 minutes, how much of 100g sample remains after 1 hour? Half-life: A(t)=A0(21)t/h (h is half-life), A(60)=100(21)60/20=100(81)=12.5g
Compound interest calculations using exponential growth model (A = P(1 + r/n)^(nt), where A is final amount, P is principal, r is annual interest rate, n is number of times compounded per year, and t is time in years)