Logarithmic Functions: math practice, Grade 11 – download the PDF

1 Logarithmic form ★★★

Rewrite each exponential equation in logarithmic form.

1. \(2^6=64\)
2. \(10^{-2}=0.01\)
3. \(7^2=49\)
4. \(5^0=1\)
5. \(9^{1/2}=3\)

2 Exponential form ★★★

Rewrite each logarithmic equation in exponential form.

1. \(\log_3 81=4\)
2. \(\log_5 125=3\)
3. \(\log_2\tfrac18=-3\)
4. \(\log_{16}4=\tfrac12\)
5. \(\log 1000=3\)

3 Evaluating logarithms ★★★

Evaluate without a calculator.

1. \(\log_2 16\)
2. \(\log_5\tfrac1{25}\)
3. \(\log_7 7\)
4. \(\log_9 1\)
5. \(\log_4 2\)
6. \(\log_{1/2}8\)

4 True or false? ★★★

Decide whether each statement is true or false and justify your answer.

1. \(\log_2 0=0\)
2. \(\log_3(-9)\) is undefined.
3. \(\log_4 4^7=7\)
4. \(\log 100=2\)
5. \(\log_2 8+\log_2 4=\log_2 12\)

5 Domain of a logarithm ★★★

Find the domain of each function.

1. \(f(x)=\log(x-4)\)
2. \(g(x)=\log_3(2x+10)\)
3. \(h(x)=\log_5(8-x)\)

6 Splitting products ★★★

Use the properties of logarithms to expand.

1. \(\log_2(8x)\)
2. \(\log(1000y^2)\)
3. \(\log_5\dfrac{x^3}{25}\)

7 Natural logarithm values ★★★

Evaluate without a calculator.

1. \(\ln e^4\)
2. \(e^{\ln 9}\)
3. \(\ln 1\)
4. \(\ln\dfrac1e\)
5. \(\ln\sqrt e\)

8 Condensing logarithms ★★★

Write each expression as a single logarithm, then simplify when possible.

1. \(\log_3 4+\log_3 5\)
2. \(\log_2 48-\log_2 3\)
3. \(2\log 5+\log 4\)
4. \(\tfrac12\ln 81-\ln 3\)

9 Change of base ★★★

Use the change-of-base formula to approximate each value to three decimal places.

1. \(\log_3 20\)
2. \(\log_7 50\)
3. \(\log_{0.5}10\)

10 Same base ★★★

Solve each equation by writing both sides as powers of the same base.

1. \(5^x=125\)
2. \(4^{x+1}=64\)
3. \(2^{3x-1}=\tfrac1{32}\)
4. \(9^x=27\)

11 Simple logarithmic equations ★★★

Solve each equation.

1. \(\log_2(x+3)=5\)
2. \(\log_5(2x-1)=2\)
3. \(\log x=-2\)
4. \(\ln x=3\) (give the exact value and a decimal approximation)

12 Reading a logarithmic graph ★★★

Consider \(f(x)=\log_2(x+5)-3\).

1. Give its domain and the equation of its vertical asymptote.
2. Find the x-intercept.
3. Find the y-intercept to three decimal places.
4. Sketch the graph using three points.

13 Using a graph and logs ★★★

The graph below shows \(y=3^x\) and the line \(y=20\).

1. Estimate from the graph the value of \(x\) for which \(3^x=20\).
2. Find that value exactly with a logarithm, and then to three decimals.

14 Growing bacteria ★★★

A culture starts with 500 bacteria and doubles every 3 hours, so \(N(t)=500\cdot 2^{t/3}\) where \(t\) is in hours.

1. When will the culture reach 8,000 bacteria?
2. When will it reach 10,000 bacteria? Round to the nearest tenth of an hour.

15 A product of logs ★★★

Solve \(\log_2 x+\log_2(x-6)=4\). Explain why one candidate must be rejected.

16 A quotient of logs ★★★

Solve \(\log(x+3)-\log(x-1)=1\).

17 Saving for a goal ★★★

You deposit $4,000 in an account. How long will it take to reach $6,000 if the account pays

1. 3.6% interest compounded annually, using \(A=4000(1.036)^t\)?
2. 3.6% interest compounded continuously, using \(A=4000e^{0.036t}\)?

Give both answers in years to the nearest hundredth and compare them.

18 Radioactive decay ★★★

An 80 mg sample of a substance has a half-life of 12 days: the amount left after \(t\) days is \(A(t)=80\left(\tfrac12\right)^{t/12}\).

1. When will 5 mg remain?
2. When will 20 mg remain?
3. When will 30 mg remain? Round to the nearest tenth of a day.

19 A hidden quadratic ★★★

Solve \(e^{2x}-5e^{x}+6=0\). Give exact values and approximations to three decimal places.

20 Finding an inverse ★★★

Let \(f(x)=2\cdot 3^{x}+1\).

1. Find a formula for \(f^{-1}(x)\).
2. State the domain of \(f^{-1}\).
3. Verify that \(f^{-1}(55)=3\).

21 Reciprocal logarithms ★★★

1. Show that \(\log_a b\cdot\log_b a=1\) for \(a,b\gt 0\), \(a,b\neq 1\).
2. Use the change-of-base formula to compute \(\log_2 9\cdot\log_3 8\).