Exponential equations and inequalities

Exponential and logarithmic functions, grade 12 (4 units) Lesson 4 of 20Learning

Exponential equations and inequalities

The same base on both sides

An exponential function is increasing or decreasing, so it takes each value once: two powers of the same base are equal only when their exponents are.

\displaystyle a^{f(x)}=a^{g(x)}\iff f(x)=g(x)

\displaystyle a>0,\ a\neq1

$\displaystyle {2^{x+1}}={32}$ : $\displaystyle {32}={2^5}$ , so $\displaystyle {2^{x+1}}={2^5}$ , $\displaystyle {x+1}={5}$ and $\displaystyle {x}={4}$ .
$\displaystyle {9^x}={27}$ : both are powers of 3, so $\displaystyle {3^{2x}}={3^3}$ , $\displaystyle {2x}={3}$ and $\displaystyle {x}={\frac{3}{2}}$ .

A common factor

The equation $\displaystyle {3^{x+2}-3^x}={72}$ has two powers of 3 whose exponents differ by a number: $\displaystyle {3^{x+2}}={9\cdot3^x}$ , so $\displaystyle 3^x$ is a common factor.

3^x(9-1)=72

Divide by 8: $\displaystyle {3^x}={9}={3^2}$ .
So $\displaystyle {x}={2}$ .

The substitution

\displaystyle {t}={a^x}

In $\displaystyle {4^x-6\cdot2^x+8}={0}$ , $\displaystyle {4^x}={(2^2)^x}={(2^x)^2}$ , so with $\displaystyle {t}={2^x}$ the equation is quadratic:

t^2-6t+8=0

$\displaystyle {t}={2}$ or $\displaystyle {t}={4}$ .
$\displaystyle {2^x}={2}$ gives $\displaystyle {x}={1}$ , and $\displaystyle {2^x}={4}$ gives $\displaystyle {x}={2}$ .

A value of $\displaystyle t$ that is not positive gives no $\displaystyle x$ , since $\displaystyle a^x>0$ for every $\displaystyle x$ . In $\displaystyle {9^x-3^{x+1}-54}={0}$ , with $\displaystyle {t}={3^x}$ : $\displaystyle {t^2-3t-54}={0}$ , so $\displaystyle {t}={9}$ or $\displaystyle {t}={-6}$ .

$\displaystyle {3^x}={9}$ gives $\displaystyle {x}={2}$ .
$\displaystyle {3^x}={-6}$ has no solution.

Exponential inequalities

An increasing function keeps the order of the numbers it is given, and a decreasing one reverses it:

When $\displaystyle a>1$ : $\displaystyle a^{f(x)}>a^{g(x)}$ exactly when $\displaystyle f(x)>g(x)$ .
When $\displaystyle 0<a<1$ : $\displaystyle a^{f(x)}>a^{g(x)}$ exactly when $\displaystyle f(x)<g(x)$ – the sign turns over.

The examples below write both sides as powers of one base:

$\displaystyle {2^{x^2-3x}<16}={2^4}$ : the base 2 keeps the sign, $\displaystyle x^2-3x<4$ , so $\displaystyle -1<x<4$ .
$\displaystyle {\left(\frac{1}{3}\right)^x>9}={\left(\frac{1}{3}\right)^{-2}}$ : the base $\displaystyle \frac{1}{3}$ turns the sign, so $\displaystyle x<-2$ .

Exercise 1

Solve.

Exercise 2

Solve by taking out a common factor.

Exercise 3

Solve by a substitution.

Exercise 4

Solve the inequalities.

Formulas: exponential and logarithmic functions, grade 12 (4 units)

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Powers

A product of powers with the same base
$\displaystyle a^{x}\cdot a^{y}$ $\displaystyle =$ $\displaystyle a^{x+y}$
A quotient of powers with the same base
$\displaystyle \frac{a^{x}}{a^{y}}$ $\displaystyle =$ $\displaystyle a^{x-y}$
A power of a power
$\displaystyle (a^{x})^{y}$ $\displaystyle =$ $\displaystyle a^{x\cdot y}$
A power of a product
$\displaystyle (a\cdot b)^{x}$ $\displaystyle =$ $\displaystyle a^{x}\cdot b^{x}$
A power of a quotient
$\displaystyle \left(\frac{a}{b}\right)^{x}$ $\displaystyle =$ $\displaystyle \frac{a^{x}}{b^{x}}$
A negative exponent
$\displaystyle a^{-x}$ $\displaystyle =$ $\displaystyle \frac{1}{a^{x}}$
The exponent 0
$\displaystyle a^{0}$ $\displaystyle =$ $\displaystyle 1$ $\displaystyle a\neq0$
A rational exponent
$\displaystyle a^{\frac{m}{n}}$ $\displaystyle =$ $\displaystyle \sqrt[n]{a^{m}}$ $\displaystyle a>0$

Logarithms (within the domain)

The definition of the logarithm
$\displaystyle \log_{a}x=b\iff a^{b}=x$ $\displaystyle a>0,\ a\neq1,\ x>0$
$\displaystyle \log_{a}(a^{b})$ $\displaystyle =$ $\displaystyle b$
$\displaystyle a^{\log_{a}x}$ $\displaystyle =$ $\displaystyle x$
The logarithm of a product
$\displaystyle \log_{a}x+\log_{a}y$ $\displaystyle =$ $\displaystyle \log_{a}(x\cdot y)$
The logarithm of a quotient
$\displaystyle \log_{a}x-\log_{a}y$ $\displaystyle =$ $\displaystyle \log_{a}\left(\frac{x}{y}\right)$
The logarithm of a power
$\displaystyle \log_{a}(x^{b})$ $\displaystyle =$ $\displaystyle b\cdot\log_{a}x$
The natural logarithm – to the base $\displaystyle e$
$\displaystyle \ln x=\log_{e}x$

Growth and decay

The amount after $\displaystyle t$ units of time
$\displaystyle f(t)=f(0)\cdot q^{t}$
The growth / decay factor – $\displaystyle p$ the percent per unit of time
$\displaystyle q=1\pm\frac{p}{100}$
The half-life $\displaystyle T$
$\displaystyle q^{T}$ $\displaystyle =$ $\displaystyle \frac{1}{2}$

Derivatives

$\displaystyle (e^{x})^{\prime}$ $\displaystyle =$ $\displaystyle e^{x}$
$\displaystyle (\ln x)^{\prime}$ $\displaystyle =$ $\displaystyle \frac{1}{x}$
The derivative of $\displaystyle e^{f(x)}$
$\displaystyle \left[e^{f(x)}\right]^{\prime}$ $\displaystyle =$ $\displaystyle e^{f(x)}\cdot f^{\prime}(x)$
The derivative of $\displaystyle \ln(f(x))$
$\displaystyle \left[\ln(f(x))\right]^{\prime}$ $\displaystyle =$ $\displaystyle \frac{f^{\prime}(x)}{f(x)}$
The derivative of a product
$\displaystyle \left[f(x)\cdot g(x)\right]^{\prime}$ $\displaystyle =$ $\displaystyle f^{\prime}(x)\cdot g(x)+f(x)\cdot g^{\prime}(x)$
The derivative of a quotient
$\displaystyle \left[\frac{f(x)}{g(x)}\right]^{\prime}$ $\displaystyle =$ $\displaystyle \frac{f^{\prime}(x)\cdot g(x)-f(x)\cdot g^{\prime}(x)}{\left[g(x)\right]^{2}}$
The derivative of a composite function
$\displaystyle \left[f(u(x))\right]^{\prime}$ $\displaystyle =$ $\displaystyle f^{\prime}(u)\cdot u^{\prime}(x)$
The tangent’s equation at the point where $\displaystyle {x}={a}$
$\displaystyle y-f(a)=f^{\prime}(a)(x-a)$

Integrals and the area

$\displaystyle \int e^{x}\,dx=e^{x}+C$
$\displaystyle \int e^{mx+b}\,dx=\frac{e^{mx+b}}{m}+C$ $\displaystyle m\neq0$
In any interval where $\displaystyle x\neq0$
$\displaystyle \int\frac{1}{x}\,dx=\ln|x|+C$
$\displaystyle \int\frac{1}{mx+b}\,dx=\frac{\ln|mx+b|}{m}+C$ $\displaystyle m\neq0$
$\displaystyle F$ an antiderivative of $\displaystyle f$
$\displaystyle \int_{a}^{b}f(x)\,dx=F(b)-F(a)$
The area between two graphs
$\displaystyle S=\int_{a}^{b}\left(f(x)-g(x)\right)dx$ $\displaystyle f(x)\geq g(x)$