Exponential equations and inequalities

Exponential and logarithmic functions, grade 12 (5 units) Lesson 2 of 23Learning

Exponential equations and inequalities

The same base on both sides

In an exponential equation the unknown is in the exponent. When both sides are powers of the same base, the exponents are equal:

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

\displaystyle a>0,\ a\neq1

$\displaystyle {9^{x}}={27}$ : $\displaystyle {3^{2x}}={3^{3}}$ , so $\displaystyle {2x}={3}$ and $\displaystyle {x}={\frac{3}{2}}$ .
$\displaystyle {2^{x}}={-4}$ has no solution: a power of a positive base is always positive.

A common factor

When the exponents differ by a constant number, take the smaller power out as a common factor. The example below solves $\displaystyle {2^{x+1}+2^{x}}={24}$ :

$\displaystyle {2^{x+1}}={2\cdot2^{x}}$ , so the left side is $\displaystyle {2\cdot2^{x}+2^{x}}={3\cdot2^{x}}$ .
$\displaystyle {3\cdot2^{x}}={24}$ , so $\displaystyle {2^{x}}={8}={2^{3}}$ and $\displaystyle {x}={3}$ .

The substitution

\displaystyle {t}={a^x}

When an equation holds $\displaystyle a^x$ and $\displaystyle {a^{2x}}={(a^x)^2}$ , substitute $\displaystyle {t}={a^x}$ and you get a quadratic equation. The example below solves $\displaystyle {4^{x}-3\cdot2^{x+1}+8}={0}$ :

$\displaystyle {4^{x}}={(2^{x})^{2}}={t^{2}}$ , and $\displaystyle {2^{x+1}}={2t}$ .
$\displaystyle {t^{2}-6t+8}={0}$ , so $\displaystyle {t}={2}$ or $\displaystyle {t}={4}$ .
$\displaystyle {2^{x}}={2}$ gives $\displaystyle {x}={1}$ ; $\displaystyle {2^{x}}={4}$ gives $\displaystyle {x}={2}$ .
$\displaystyle {t}={a^x}$ is always positive: a negative solution for $\displaystyle t$ gives no $\displaystyle x$ .

The same exponent, different bases

Powers with the same exponent join into one power by $\displaystyle {a^x\cdot b^x}={(a\cdot b)^x}$ :

$\displaystyle {2^{x}\cdot3^{x}}={36}$ : $\displaystyle {6^{x}}={6^{2}}$ , so $\displaystyle {x}={2}$ .
$\displaystyle {3^{x}}={5^{x}}$ : divide by $\displaystyle 5^x$ and you get $\displaystyle {\left(\frac{3}{5}\right)^{x}}={1}$ , so $\displaystyle {x}={0}$ .

Exponential inequalities

$\displaystyle a^x$ increases when $\displaystyle a>1$ and decreases when $\displaystyle 0<a<1$ . So when you compare the exponents:

for $\displaystyle a>1$ the sign stays: $\displaystyle 2^{x}>2^{5}$ means $\displaystyle x>5$ ;
for $\displaystyle 0<a<1$ the sign turns: $\displaystyle \left(\frac{1}{3}\right)^{x}>\left(\frac{1}{3}\right)^{2}$ means $\displaystyle x<2$ .

Worked example: an inequality by substitution

The example below solves $\displaystyle 9^{x}-10\cdot3^{x}+9<0$ with the substitution $\displaystyle {t}={3^x}$ :

$\displaystyle t^{2}-10t+9<0$ , that is $\displaystyle (t-1)(t-9)<0$ , so $\displaystyle 1<t<9$ .
$\displaystyle 1<3^{x}<9$ is $\displaystyle 3^{0}<3^{x}<3^{2}$ , and the base 3 is greater than 1.
The solution: $\displaystyle 0<x<2$ .

Exercise 1

Solve.

Exercise 2

Solve with a common factor.

Exercise 3

Solve with a substitution.

Exercise 4

Solve the inequalities.

Formulas: exponential and logarithmic functions, grade 12 (5 units)

Open as a page

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
$\displaystyle \log_{a}(x^{b})$ $\displaystyle =$ $\displaystyle b\cdot\log_{a}x$
Change of base
$\displaystyle \log_{c}x$ $\displaystyle =$ $\displaystyle \frac{\log_{a}x}{\log_{a}c}$
$\displaystyle \log_{a}b$ $\displaystyle =$ $\displaystyle \frac{1}{\log_{b}a}$
The natural logarithm – to the base $\displaystyle e$
$\displaystyle \ln x=\log_{e}x$

The exponential and the logarithmic function

Every exponential as a power of $\displaystyle e$
$\displaystyle a^{x}$ $\displaystyle =$ $\displaystyle e^{x\ln a}$
$\displaystyle a^x$ and $\displaystyle \log_a x$ – inverse functions
$\displaystyle y=a^{x}\iff x=\log_{a}y$ $\displaystyle y>0$
The number $\displaystyle e$
$\displaystyle e\approx2.718$

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 $\displaystyle T$
$\displaystyle q^{T}$ $\displaystyle =$ $\displaystyle \frac{1}{2}$
The doubling time $\displaystyle T$
$\displaystyle q^{T}$ $\displaystyle =$ $\displaystyle 2$

Derivatives

$\displaystyle (e^{x})^{\prime}$ $\displaystyle =$ $\displaystyle e^{x}$
$\displaystyle (a^{x})^{\prime}$ $\displaystyle =$ $\displaystyle a^{x}\cdot\ln a$
$\displaystyle (\ln x)^{\prime}$ $\displaystyle =$ $\displaystyle \frac{1}{x}$
$\displaystyle (\log_{a}x)^{\prime}$ $\displaystyle =$ $\displaystyle \frac{1}{x\cdot\ln a}$
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, the area and accumulation

$\displaystyle \int e^{x}\,dx=e^{x}+C$
$\displaystyle \int a^{x}\,dx=\frac{a^{x}}{\ln a}+C$
In any interval where $\displaystyle x\neq0$
$\displaystyle \int\frac{1}{x}\,dx=\ln|x|+C$
$\displaystyle F$ an antiderivative of $\displaystyle f$
$\displaystyle \int f(mx+b)\,dx=\frac{1}{m}F(mx+b)+C$ $\displaystyle m\neq0$
$\displaystyle F$ an antiderivative of $\displaystyle f$
$\displaystyle \int f(u(x))\cdot u^{\prime}(x)\,dx=F(u(x))+C$
Where $\displaystyle f(x)\neq0$
$\displaystyle \int\frac{f^{\prime}(x)}{f(x)}\,dx=\ln|f(x)|+C$
$\displaystyle F$ an antiderivative of $\displaystyle f$
$\displaystyle \int_{a}^{b}f(x)\,dx=F(b)-F(a)$
The area between two graphs
$\displaystyle S=\int_{a}^{b}\left(f(x)-g(x)\right)dx$ $\displaystyle f(x)\geq g(x)$
The accumulation function’s derivative
$\displaystyle \left(\int_{a}^{x}f(t)\,dt\right)^{\prime}=f(x)$