Precalculus Exponential Log Lesson: Laws, Models

Contents

Key takeaway: This precalculus exponential log lesson starts from one idea: a logarithm is an exponent. Writing $\log_{b}x=y$ means that $b^{y}=x$, so exponential and logarithmic functions undo each other and their graphs mirror across the line $y=x$. The lesson then builds the laws of logarithms, solves equations with unknown exponents and applies the tools to money, populations and sound. Standard F-BF.B.5 supports the inverse view.

A logarithm is an exponent: log base b of x equals y exactly when b to the power y equals x. Exponential and logarithmic functions are inverses, so their graphs mirror each other in the line y = x, and the laws of logarithms turn products into sums and powers into multipliers, which lets you solve for exponents.

PrecalculusCommon Core: F-LE.A.4 · F-BF.B.5 · F-IF.C.8b

Precalculus Exponential Log Lesson: key facts

log_b x = y means b^y = x, with b > 0, b not equal to 1 and x > 0.
Product, quotient and power laws: log(MN) = log M + log N, log(M/N) = log M – log N, log(M^k) = k log M.
Solve b^t = c with t = ln c / ln b, and always check log equations for extraneous answers.
Continuous growth is A = Pe^(rt); a logistic model P = K/(1 + Ae^(-rt)) levels off at K.
Log scales (pH, decibels, magnitude): a difference of d units means a ratio of 10^d.

How are exponents and logarithms inverses?

A logarithm is an exponent. The statement $\log_{b} x=y$ means $b^{y}=x$, where $b > 0$, $b\neq 1$ and $x > 0$. Because the two functions undo each other, $\log_{b} b^{x}=x$ and $b^{\log_{b} x}=x$. Their graphs are mirror images in the line $y=x$: every point $(a,b)$ on one graph matches $(b,a)$ on the other, so domain and range swap and the horizontal asymptote $y=0$ becomes the vertical asymptote $x=0$.

The graphs of y = 2^x and y = log2 x reflect across the line y = x.

Vocabulary

Common logarithm: base 10, written $\log x$. Natural logarithm: base $e\approx 2.718$, written $\ln x$.

Carrying capacity: the limiting value $K$ that a logistic model approaches.

What are the laws of logarithms?

Law	Rule
Product	$\log_{b} (MN)=\log_{b} M+\log_{b} N$
Quotient	$\log_{b} \frac{M}{N}=\log_{b} M-\log_{b} N$
Power	$\log_{b} M^{k}=k\log_{b} M$
Change of base	$\log_{b} M=\frac{\ln M}{\ln b}$

Multiplying adds exponents, so the log of a product is a sum of logs.

How do you solve exponential and logarithmic equations?

Isolate the exponential (or the logarithm) on one side.
Take a logarithm of both sides, or rewrite the log equation in exponential form.
Solve the resulting equation, then check every answer in the original equation.

Example. A $500 deposit earns 5% interest per year, so $500(1.05)^{t}=1500$ tells when it reaches $1,500. Divide by 500: $1.05^{t}=3$. Then $t=\frac{\ln 3}{\ln 1.05}\approx 22.52$ years.

The curve y = 500(1.05)^t crosses y = 1500 at t = 22.52 years.

A logarithm accepts only positive inputs, so an answer that makes any log argument zero or negative is extraneous and must be rejected.

How do continuous growth and logistic models work?

Any growth factor $b$ can be written with base $e$: $b^{t}=e^{t\ln b}$. Continuous growth uses $A=Pe^{rt}$, and the doubling time is $\frac{\ln 2}{r}$. Real populations cannot grow forever, so a logistic model $P(t)=\frac{K}{1+Ae^{-rt}}$ levels off at the carrying capacity $K$. It grows fastest when $P=\frac{K}{2}$, at $t=\frac{\ln A}{r}$.

A logistic curve rises fastest at half the limit, then levels off at P = 1000.

How do logarithmic scales work?

On a logarithmic scale each step multiplies the quantity by 10. For acidity, $\text{pH}=-\log [H^{+}]$; for sound, $L=10\log\frac{I}{I_{0}}$ decibels; for earthquakes, one magnitude unit means 10 times the wave amplitude. A difference of $d$ units therefore means a ratio of $10^{d}$.

The pH scale is logarithmic: each whole step is a factor of 10.

Exponential and Logarithmic Functions and Models: frequently asked questions

Why are exponential and logarithmic functions inverses?

Exponentiating undoes taking a logarithm and the other way around, since log_b(b^x) = x and b^(log_b x) = x. Graphically, swapping x and y reflects one graph across the line y = x.

Why can you not take the log of a negative number?

Every output of b^y is positive, so no exponent produces a negative or zero number. The domain of log_b x is x > 0, which is why some solutions of log equations must be rejected.

How do you solve an equation with the unknown in an exponent?

Isolate the exponential part, take ln of both sides and use the power law to bring the exponent down. For 5^x = 40, x = ln 40 / ln 5, about 2.29.

What is the difference between ln and log?

The symbol log means base 10 and ln means base e, about 2.718. Both obey the same laws, and the change-of-base formula converts between them.

What does the carrying capacity K in a logistic model mean?

K is the largest value the quantity can approach, such as the number of fish a pond can support. The graph rises toward the horizontal line y = K but never crosses it.

How are exponents and logarithms inverses?

Start with $4^{3/2}=8$. In logarithmic form it reads $\log_{4}8=\frac{3}{2}$. The base must be positive and not equal to 1, and the input of a log must be positive. Therefore $\log$ of a negative number does not exist. Since the two functions are inverses, the point $(2,9)$ on $y=3^{x}$ pairs with $(9,2)$ on $y=\log_{3}x$.

What are the laws of logarithms?

The product law turns multiplication into addition: $\ln 3+\ln 4=\ln 12$. The quotient law turns division into subtraction. The power law moves an exponent to the front, so $2\log_{b}x-\log_{b}y=\log_{b}\frac{x^{2}}{y}$. However, no law exists for the log of a sum, which is a frequent trap.

How do you solve exponential and logarithmic equations?

When the unknown sits in an exponent, take a log of both sides. For $5^{x}=40$, the result is $x=\frac{\ln 40}{\ln 5}\approx 2.29$. When the unknown sits inside a log, combine the logs and rewrite in exponential form. Then test each answer, because $\ln x+\ln(x-3)=\ln 4$ leads to $x=4$ or $x=-1$, and only 4 works.

Where do continuous growth and logistic models appear?

Money growing continuously at 3% doubles when $t=\frac{\ln 2}{0.03}\approx 23.1$ years. A logistic curve levels off at a carrying capacity, which is the number its formula approaches over time.

What should you practice next?

Pick one growth story from daily life, such as a savings account or a viral video. Write its model, solve for the time to double, and check the answer by substituting back. That loop turns the laws into a dependable habit.