Logarithmic equations and inequalities

Exponential and logarithmic functions, grade 12 (4 units) Lesson 8 of 20Learning

Logarithmic equations and inequalities

The domain first

An equation with logarithms starts from its domain: every expression inside a logarithm must be positive.

At the end, every solution is checked against the domain.
A solution outside the domain is no solution of the equation, even if it solves an equation reached on the way.

One logarithm equal to a number

$\displaystyle {\log_2(x+1)}={3}$ . The domain: $\displaystyle x>-1$ . By the definition of the logarithm:

x+1=2^3

$\displaystyle {x+1}={8}$ , so $\displaystyle {x}={7}$ .
$\displaystyle 7>-1$ : the solution is in the domain.

A sum of logarithms

$\displaystyle {\log x+\log(x-3)}={1}$ . The solution below finds the domain, joins the logarithms by the law of the logarithm of a product, and goes over to the power form.

The domain: $\displaystyle x>0$ and $\displaystyle x>3$ , that is $\displaystyle x>3$ .
$\displaystyle {\log(x(x-3))}={1}$ , so $\displaystyle {x^2-3x}={10^1}$ .
$\displaystyle {x^2-3x-10}={0}$ : $\displaystyle {x}={5}$ or $\displaystyle {x}={-2}$ .
$\displaystyle -2$ is not in the domain. The solution: $\displaystyle {x}={5}$ .

A logarithm equal to a logarithm

Two different numbers cannot have the same logarithm to the same base. So when $\displaystyle {\log_a f(x)}={\log_a g(x)}$ , then $\displaystyle {f(x)}={g(x)}$ .

$\displaystyle {\log_2(x+3)}={\log_2(2x)}$ . The domain: $\displaystyle x>-3$ and $\displaystyle x>0$ , that is $\displaystyle x>0$ .
$\displaystyle {x+3}={2x}$ , so $\displaystyle {x}={3}$ , in the domain.
$\displaystyle {\log_3(x^2-2x)}={1}$ is one of these too: $\displaystyle {1}={\log_3 3}$ , so $\displaystyle {x^2-2x}={3}$ , and $\displaystyle {x}={3}$ or $\displaystyle {x}={-1}$ . For both $\displaystyle {x^2-2x}={3>0}$ , so both are solutions.

The substitution

\displaystyle {t}={\log x}

In $\displaystyle {(\log x)^2-3\log x+2}={0}$ the unknown appears only inside $\displaystyle \log x$ . We substitute $\displaystyle {t}={\log x}$ and get a quadratic equation:

t^2-3t+2=0

$\displaystyle {t}={1}$ or $\displaystyle {t}={2}$ .
$\displaystyle {\log x}={1}$ : $\displaystyle {x}={10}$ . $\displaystyle {\log x}={2}$ : $\displaystyle {x}={100}$ .
The domain $\displaystyle x>0$ : both solutions are in it.

An inequality between logarithms to the same base

Within the domain:

$\displaystyle a>1$ : the function $\displaystyle \log_a x$ increases, so $\displaystyle \log_a f(x)>\log_a g(x)$ where $\displaystyle f(x)>g(x)$ – the sign stays.
$\displaystyle 0<a<1$ : the function $\displaystyle \log_a x$ decreases, so $\displaystyle \log_a f(x)>\log_a g(x)$ where $\displaystyle f(x)<g(x)$ – the sign turns over.
A number on the other side is written as a logarithm to the same base: $\displaystyle {3}={\log_2 8}$ , $\displaystyle {-2}={\log_{\frac{1}{2}}4}$ .
The answer is the part that the solution and the domain share.

Examples of inequalities

$\displaystyle \log_2(x-1)<3$ : the domain $\displaystyle x>1$ . $\displaystyle {3}={\log_2 8}$ and the base is greater than 1, so $\displaystyle x-1<8$ , that is $\displaystyle x<9$ . With the domain: $\displaystyle 1<x<9$ .
$\displaystyle \log_{\frac{1}{2}}(x-1)>-2$ : the domain $\displaystyle x>1$ . $\displaystyle {-2}={\log_{\frac{1}{2}}4}$ and the base is less than 1, so the sign turns over: $\displaystyle x-1<4$ , that is $\displaystyle x<5$ . With the domain: $\displaystyle 1<x<5$ .
$\displaystyle \log_3(x^2-2x)<1$ : the domain $\displaystyle x^2-2x>0$ , that is $\displaystyle x<0$ or $\displaystyle x>2$ . $\displaystyle x^2-2x<3$ gives $\displaystyle -1<x<3$ . The shared part: $\displaystyle -1<x<0$ or $\displaystyle 2<x<3$ .

Exercise 1

Solve the equations.

Exercise 2

Solve the equation.

Exercise 3

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
$\displaystyle S=\int_{a}^{b}\left(f(x)-g(x)\right)dx$ $\displaystyle f(x)\geq g(x)$