Learn – Algebra/Precalculus – Logarithms

Definition

Exercises

Problem set

Without using calculators, give the values of the following.

  1. \log^{64}_4

  2. \log^{81}_3

  3. \log^{1000}_{10}

  4. \log^{7}_{7}

  5. \log^{0.2345}_{0.2345}

Problem set

Without using calculators, give the values of the following.

  1. \log^{1}_2

  2. \log^{1/4}_2

  3. \log^{1/27}_9

  4. \log^{81}_{1/27}

  5. \log^{1/32}_{16}

Problem set

Without using calculators, give the values of the following.

  1. \log^{0.1}_{0.01}

  2. \log^{100}_{0.1}

  3. \log {100000}

  4. \log {0.001}

  5. \log {1000^{2/3}}

  6. \ln {e^{2}}

Problem set

  1. Which one of \log^7_5 and \log^8_5 is bigger?

  2. Which one of \log^7_2 and \log^7_3 is bigger?

  3. Which one of e^{-2} and e^{-3} is bigger?

Logarithmic identities

Exercises

Problem set

Justify each of the following logarithmic identities.

  1. \log^{a}_a = 1

  2. \log^{1}_a = 0

  3. \log^{xy}_a = \log^x_a+\log^y_a

  4. \log^{\frac{x}{y}}_a = \log^x_a-\log^y_a

  5. \log^{x^m}_a = m\log^{x}_a

  6. \log_{a^m}^x = \frac{1}{m}\log_a^x

  7. a^{\log^{x}_a} = x

  8. \log^x_b\times\log^b_a = \log^x_a

  9. \frac{\log^x_b}{\log^a_b} = \log^x_a

  10. \frac{1}{\log^b_a} = \log^a_b

  11. x^{\log^{y}_a} = y^{\log^{x}_a}

Problem set

Without using calculators, give the values of the following.

  1. \log^{2^{100}}_2

  2. \log^{11^8}_{11}

  3. \log^{4^{100}}_2

  4. 2^{\log^5_2}

  5. 4^{\log^9_4}

  6. 7^{\log^{14}_7}

  7. e^{\ln 3}

Problem set

Evaluate the following.

  1. \log^9_6 + \log^4_6

  2. \log^{16}_{20} + \log^{25}_{20}

  3. \log^{12}_2 - \log^3_2

  4. \log^{54}_3 - \log^2_3

  5. \log_5^{123} + \log_5^{1/123}

Problem set

Evaluate the following.

  1. \log^4_{10} + \log^{50}_{10} - \log^2_{10}

  2. \log^{0.006}_{10} - \log^2_{10} - \log^3_{10}

  3. \log^{20}_{10} + \log^{0.1}_{10} - \log^2_{10}

Problem set

Evaluate the following.

  1. \log^{8^{51}}_2

  2. \log^{49^{11}}_7

  3. \log^5_{25^3}

  4. \log^{9^{51}}_{27}

  5. \log^{8^{25}}_{16}

Problem set

Evaluate the following.

  1. \log^{0.1^{33}}_{10}-\log^{0.001^{11}}_{10}

  2. \log^{9^{-24}}_6 + \log^{4^{-24}}_6

  3. \log^{8}_{2^{24}}

  4. \log^{9}_{27^{27}}

  5. \log^{25^{31}}_{5^{31}}

Problem set

Evaluate the following.

  1. 7^{\log^{11}_{7}}

  2. 5^{\log^{23}_{5}}

  3. 5^{\log^{33}_5-\log^{11}_5}

  4. 8^{5\log^{21}_8}

  5. 11^{-\log^{10}_{11}}

  6. 3^{\log^{8}_{9}}

  7. 5^{\log^{27}_{25}}

  8. 2^{-4\log^{3}_{4}}

Problem set

Evaluate the following.

  1. \log^{16}_3\times \log^3_2

  2. \log^{125}_9\times \log^{81}_5

  3. \log^5_3\times \log^{3^5}_{125}

  4. \frac{\log^{27}_{5}}{\log^{9}_{5}}

  5. \frac{\log^{a^3}_9}{\log^{a^7}_9}

  6. \frac{\log^{2}_a}{\log^{8}_{a^6}}

  7. \frac{\log^{10}_p}{\log^{100}_{p^2}}

Problem set

For each of the following, answer if the equation is true or false.

  1. \log^x_a\times\log^y_a = \log^{xy}_a

  2. \log^{(x+y)}_a = \log^x_a\times\log^y_a

  3. \frac{\log^x_a}{\log^y_a} = \log^{(x-y)}_a

  4. \log {p!} = \log {p}+\log {(p-1)}+\cdots+\log 3 + \log 2+\log 1

  5. \log^a_{b^n} = n\times\log^a_b

Problem set

For each of the following, answer if the equation is true or false.

  1. 3^{5\log^3_3} = 5

  2. 4^{5\log^7_4} = 8^{5\log^{7}_8}

  3. 9^{\log^5_{1/9}} = \frac{1}{5}

  4. 6^{5\log^2_6} = 2^5

  5. x^x = 4^{x\log^x_4}

  6. x^{\log^x_4} = 4^{(\log^x_4)^2}

Problem set

For each of the following, answer if the equation is true or false.

  1. 5^{xy} = 4^{x\log^{5^y}_4}

  2. \log^{\log^5_2}_2 = (\log^5_2)^2

  3. \log^{3^{5\ln 5}}_3 = 5\ln 5

  4. \log^{2^{5}}_8 = 5

  5. 9^{\log^5_2} = 3^{\log^{25}_2}

Problem set

For each of the following, answer if the equation is true or false.

  1. -\log^{1/a}_{b} = \log^a_b

  2. \log^{1/a}_{b} = \log^a_{1/b}

  3. a^{\log^x_a} = b^{\log^x_b}

  4. a^{-\log^x_a} = -x

  5. a^{\log^x_{1/a}} = -x

Problem set

For each of the following, answer if the equation is true or false.

  1. \log^{\left(\frac{1}{x}\right)}_{a} = \log^x_{\left(\frac{1}{a}\right)}

  2. -\log^x_{a} = \log^x_{\left(\frac{1}{a}\right)}

  3. \log^{x^m}_{a} = \log^x_{a^m}

  4. \log^{x^m}_{a} = -\log^x_{a^m}

  5. \log^{x^m}_{a} = \log^x_{\left(a^{1/m}\right)}

Problem set

For each of the following, answer if the equation is true or false.

  1. \frac{\log^x_a}{\log^y_a} = \log^{\left(\frac{x}{y}\right)}_a

  2. \frac{\log^m_a}{\log^n_a} = \log^m_n

  3. \log^{a}_{b}\times \log^b_{a} = 1

  4. \frac{\log^{\left(a^2\right)}_{b}}{\log^a_{\left(b^2\right)}} = 1

  5. e^{3\ln 11} = 8^{3\frac{\ln 11}{\ln 8}}

  6. \frac{1}{\log^{a}_{b}} = \log^{1/a}_b

  7. \log^b_a = \frac{1}{\log^{a}_{b}}

Logarithmic functions

Exercises

Problem set

Graph and identify domain and range for each of the following functions.

  1. \log^x_2

  2. \log^x_{9}

  3. \log^x_{7}

  4. \log^{(x-5)}_5

  5. -\log^{x}_9

  6. \log^{-x}_9

Problem set

Graph and identify domain and range for each of the following functions.

  1. -\log^{x+7}_8

  2. \log^{(x-3)}_{0.2}

  3. \log^x_{0.5}

  4. \ln x

  5. \ln (x+1)

Miscellaneous

Exercises

Problem set

Evaluate the following.

  1. (\log^2_6)^2+2\log^2_6\log^3_6 + (\log^3_6)^2

  2. (\log 7)^2-2\log 7\log {700} + (\log 700)^2

Problem set

  1. What is the value of x such that a^x = -a?
  2. What is the value of x such that a^x = \frac{a}{a^2}?
  3. What is the value of x such that -{0.1}^x = \frac{1}{10}?
  4. Solve for x: 2^{2x} - 5\times 2^x+4 = 0. Hint: Substitute y = 2^x.
  5. Solve for x: 4\times2^{2x} - 5\times 2^x = -1.

Problem set

For the following knowing the graphs of f(x), plot the graphs of g(x) using translation, scaling and reflection transformations, or inverse function relationships.

  1. f(x) = a^x, g(x) = \left(\frac{1}{a}\right)^x for a > 1
  2. f(x) = 2^x, g(x) = \left(\sqrt{2}\right)^x
  3. f(x) = 2^x, g(x) = 2\times2^x+2
  4. f(x) = 5^x, g(x) = 0.2^x
  5. f(x) = (0.2)^x, g(x) = \log^x_{0.2}
  6. f(x) = \log^x_2, g(x) = \log^{(x+1)^3}_8

Problem set

Solve for x in the following.

  1. \log^{100000}_{10} = 10\ln x
  2. \log^{a}_{\sqrt[4]{a}} = \frac{x}{8}
  3. \log^a_2 = x\ln a
  4. e^{2\ln x} = 1
  5. \frac{1}{\log^x_2} = \ln 4

Problem set

Solve for x in the following.

  1. Solve for x: \ln(x-1) + \ln(2x-1) = \ln 2+\ln(2x^2+1)
  2. Solve for x: e^{\ln(x+1) - \ln(x-1)} = 2

Problem set

  1. Determine the x and y intercepts of \ln (x+e).
  2. \log^a_2 = \log^2_a, and a < 1. What is the value of a?
  3. 1+\log^a_b = \log^{1+a}_b. Express a in terms of b.
  4. e^{\log^4_a} = 2. What is the value of a?
  5. e^{ax} = 2^x. What is the value of a?
  6. 3^p = 2^r. Express r in terms of p.
  7. If 9^{ax} = e^x, what is the value of a?
  8. If 5^{t} = e^{x}, express t in terms of x.