Logarithmic functions are powerful tools for understanding exponential relationships. They convert multiplication to addition, making complex calculations simpler. This topic explores how to work with logs, convert between forms, and apply them to real-world scenarios.
Logs pop up in various fields, from measuring earthquakes to calculating compound interest. By mastering logarithms, you'll gain insights into growth, decay, and scale in nature and science. These skills are crucial for tackling advanced math and real-life problems.
Logarithmic Functions
Logarithmic and exponential form conversion
Logarithmic form logb(x)=y equivalent to exponential form by=x
b represents the base, x represents the argument, y represents the
Converting logarithmic to exponential form:
Base of logarithm becomes base of exponent
Logarithm becomes the exponent
Argument becomes result of exponential expression (x=by)
Converting exponential to logarithmic form:
Base of exponent becomes base of logarithm
Exponent becomes the logarithm
Result of exponential expression becomes argument (y=logb(x))
The inverse operation of a logarithm is called the antilogarithm
Evaluation of logarithms with bases
Common logarithms have base 10, denoted as log(x)
log(100)=2 since 102=100
Natural logarithms have base e (Euler's number, ~2.71828), denoted as ln(x)
ln(e3)=3 since e3≈20.0855
Evaluating logarithms with other bases uses change of base formula:
logb(x)=loga(b)loga(x), where a is any base (often 10 or e)
Power rule: logb(xn)=nlogb(x) (log4(43)=3log4(4)=3⋅1=3)
Common logarithms in problem-solving
measures earthquake magnitudes
Magnitude M=log(I0I), I is earthquake intensity, I0 is reference intensity
Magnitude 5 earthquake has intensity 10 times greater than magnitude 4 (105/104=10)
Decibel scale measures sound intensity
Decibels β=10log(I0I), I is sound intensity, I0 is reference intensity
60 dB sound has intensity 1,000,000 times greater than 0 dB (106=1,000,000)
pH scale measures acidity of solutions
pH=−log([H+]), [H+] is hydrogen ion concentration
Solution with pH 3 has 10 times more H+ ions than solution with pH 4 (10−3/10−4=10)
These are examples of logarithmic scales, which compress wide ranges of values
Natural logarithms for growth and decay
Exponential growth: A(t)=A0ekt, A(t) is amount at time t, A0 is initial amount, k is growth rate
Solving for time: t=kln(A0A(t))
Population doubles in 10 years with 7% annual growth rate (0.07ln(2)≈9.9 years)
Exponential decay: A(t)=A0e−kt, A(t) is amount at time t, A0 is initial amount, k is decay rate
Solving for time: t=−kln(A0A(t))
Radioactive isotope decays to 25% of original amount in 2 half-lives (2⋅kln(2))
Half-life is time for quantity to reduce to half its initial value
Half-life formula: t1/2=kln(2), k is decay rate
Carbon-14 has half-life of 5,730 years (0.000121ln(2))
Interpretation of logarithmic expressions
Logarithm represents power base must be raised to obtain the number
log2(8)=3 means 23=8
In exponential growth/decay, ln(initialcurrent)/rate gives elapsed time
ln(5001000)/0.05≈13.9 years for 500togrowto1000 at 5% rate
Logarithmic scales compress wide range of values
Each unit increase represents tenfold increase in measured quantity
pH 6 is 10 times more acidic than pH 7, 100 times more than pH 8
Graphing Logarithmic Functions
Domain of logarithmic functions is all positive real numbers (x > 0)
Range of logarithmic functions is all real numbers
Logarithmic functions have a vertical asymptote at x = 0
When graphing, consider the base of the logarithm and any transformations applied
Key Terms to Review (4)
Common logarithm: A common logarithm is a logarithm with base 10. It is often written as $\log_{10} x$ or simply $\log x$.
Logarithm: A logarithm is the inverse operation to exponentiation, meaning it undoes the process of raising a number to a power. The logarithm of a number $x$ with base $b$ is the exponent to which $b$ must be raised to produce $x$.
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.
Richter Scale: The Richter Scale measures the magnitude of earthquakes using a logarithmic scale. Each whole number increase on the scale represents a tenfold increase in measured amplitude and roughly 31.6 times more energy release.