Review: exponential and logarithmic functions

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

Review: exponential and logarithmic functions

Solve. With a logarithm – mind the domain.

A patient’s blood holds 4 mg of a medicine.
The medicine’s half-life in the blood is 3 hours.

Work out the decay factor per hour. Round to four decimal places.

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How many mg of the medicine will the blood hold after 6 hours?

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A blood test finds the medicine while the blood holds at least 0.1 mg of it.
After how many hours will the test no longer find it? Solve the equation, then work out the time and round it to two decimal places.

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The function $\displaystyle {f(x)}={(x^2-3)e^x}$ is given.

Find $\displaystyle f'(x)$ . Take out a common factor, and factor.

$\displaystyle f(x)=(x^{2}-3)e^{x}$

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Find the extremum points of the function, and the type of each.

$\displaystyle f(x)=(x^{2}-3)e^{x}$

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$\displaystyle f(x)=(x^{2}-3)e^{x}$

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Find $\displaystyle f''(x)$ .

$\displaystyle f(x)=(x^{2}-3)e^{x}$

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Find the x-coordinates of the inflection points: solve the equation $\displaystyle {f''(x)}={0}$ .

Where $\displaystyle x$ is negative and far from 0, $\displaystyle (x^2-3)e^x$ is close to 0, so $\displaystyle {y}={0}$ is a horizontal asymptote on the left.
Mark the extremum points and the inflection points on the graph.

\displaystyle f(x)=(x^{2}-3)e^{x}

\displaystyle y=0

The function $\displaystyle {f(x)}={\ln(x^2-4x+5)}$ is given.

Explain why the function is defined for every $\displaystyle x$ .
Without a derivative: find the extremum point of the function and its type. Use the parabola $\displaystyle {y}={x^2-4x+5}$ .

Find $\displaystyle f'(x)$ .

$\displaystyle f(x)=\ln(x^{2}-4x+5)$

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$\displaystyle {f''(x)}={\frac{-2x^2+8x-6}{(x^2-4x+5)^2}}$ .
Find where the function is concave up: solve the inequality $\displaystyle f''(x)>0$ .

Mark the extremum point and the inflection points on the graph.

\displaystyle f(x)=\ln(x^{2}-4x+5)

The drawing shows the graphs of $\displaystyle {f(x)}={e^{2-x}}$ and $\displaystyle {g(x)}={e^x}$ .
The shaded area is bounded by the two graphs and the y-axis.

\displaystyle f(x)=e^{2-x}

\displaystyle g(x)=e^{x}

Find the x-coordinate of the point where the two graphs meet.

Work out the shaded area: write the integral and work it out.

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Integrals: a function from its derivative, and an amount that accumulates.

The derivative of the function $\displaystyle f$ is $\displaystyle {f'(x)}={\frac{2x}{x^2+1}+e^{-x}}$ .
The graph of $\displaystyle f$ passes through the origin, $\displaystyle (0,0)$ .
Work out the integral, then find $\displaystyle C$ and write $\displaystyle f(x)$ .

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Water flows into a pool at the rate $\displaystyle {r(t)}={50e^{-0.1t}}$ litres a minute, $\displaystyle t$ minutes after the tap is opened.
How many litres flowed into the pool in the first 10 minutes? Work out the exact value, then round to two decimal places.

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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
$\displaystyle \left(\int_{a}^{x}f(t)\,dt\right)^{\prime}=f(x)$