Simple logarithmic equations: 1, 2, 3

More difficult logarithmic equations: 4, 5

Exponential equations: 6, 7, 8, 9, 10

Overview of the procedures

All tasks on one page for printing

TOP | Task 1 | Solve the following logarithmic equations by putting them in the form loga = b and then raising them to the power of 10. | a) | log (x-5) = -2 | d) | log (7x + 9) - logx = 1 | b) | log (3x-2) = 1 | e) | 2logx - log (4x-3) = 0 | c) | log (2x) + log4 = 3 | f) | logx + log (x + 2) - log3 = 0 |
| | SOLUTION |

TOP | exercise 2 | Solve the following logarithmic equations by putting them in the form loga = logb and then raising them to the power of 10. | a) | logx + log (x-7) = log6 + log3 | d) | log (x-5) - log2 = log (3x) | b) | log (x-3) - log6 = log7 - log (x-4) | e) | 2log (x + 1) - logx = log4 | c) | log (35-x^{3}) = 3log (5-x) | f) | 4logx = 2log (x^{2}-3x) |
| | SOLUTION |

TOP | Task 3 | Two tasks with a different basis. Try to solve them using the same pattern: | a) | log_{2}(x + 14) - log_{2}(2x) = 2 | b) | log_{2}(x + 3) + log_{2}(x-2) = 1 + log_{2}x |
| | SOLUTION |

TOP | Task 4 | First, solve the quadratic equation: | a) | 2 (logx)^{2}-5logx-3 = 0 | b) | (log_{2}(x))^{2} - 7log_{2}(x) + 12 = 0 |
| | SOLUTION |

TOP | Task 5 | The following tasks can be solved if you take both sides logarithmically: | a) | x^{1 + logx} = 10^{2} | b) | x^{3} = 10x^{1 + logx} |
| | SOLUTION |

TOP | Exercise 6 | Six simple exponential equations that can be solved without logarithms: | a) | 5^{x} = 15'625 | b) | 2^{2x} = 64 | c) | 10^{x} = 100^{-1.5} | d) | | e) | | f) | |
| | SOLUTION |

TOP | Exercise 7 | Four more difficult exponential equations that can be solved without logarithms: | a) | 2^{3x-4}^{ . }4^{2x-3} = 8^{x + 2} | c) | 7^{x}+8^{ . }7^{x-1} = 735 | b) | 3^{4x-1}^{ . }9^{2x + 1} = 27^{x. }3^{5x + 1} | d) | 8^{2x-1} - 4^{3x-1} + 2^{6x-1} = 96 |
| | SOLUTION |

TOP | Exercise 8 | Six problems that lead to quadratic equations: | a) | 3^{2x} - 12^{ . }3^{x} + 27 = 0 | d) | 4^{x + 1} - 2^{x + 4} = 128 | b) | 4^{x} - 12^{ . }2^{x} + 32 = 0 | e) | 25^{x + 1} + 3^{ . }5^{x + 2} - 16 = 0 | c) | 2^{2x-1} - 3^{ . }2^{x} + 4 = 0 | f) | 4^{x + 1} + 16^{x-1} = 1536 |
| | SOLUTION |

TOP | Exercise 9 | Six simple exponential equations that require logarithmizing both sides of the equation: | a) | 2^{3x} = 5 | b) | 5^{3x-2} = 7 | c) | 2^{x-2} = 5^{x-1} | d) | | e) | | f) | 3^{x-1}^{ . }2^{2x} = 5^{3x + 1} |
| | SOLUTION |

TOP | Exercise 10 | Six more difficult exponential equations that sooner or later have to use logarithms: | a) | 2^{x-1}^{ . }5^{2x-1} = 3^{1-x} | d) | 4^{x-1} - 9^{x} = 3^{2x-1} - 2^{2x + 1} | b) | 5^{x} + 6^{x} = 6^{x + 1} | e) | 2^{2}^{.} 5^{x} - 2^{2x} = 2^{2x + 2} | c) | 2^{x + 1} - 3^{x} = 3^{x-1} - 2^{x} | f) | 2^{4x} + 2^{4x + 5} = 99 |
| | SOLUTION |