How to Solve Logarithmic Problems: Essential Tips for Students

In this video, I solve a variety of questions based on logarithmic concepts, covering different types of problems that middle and high school students often encounter in exams. I provide step-by-step explanations to help you approach and solve various logarithmic problems effectively.

Since the equation 1x = b has no solutions when b ≠ 1 and has infinitely many solutions when b = 1, it does not provide a well-defined, unique value for x. Therefore, using 1 as a base in a logarithm is problematic, and as a result, log1(b) is undefined. The base of a logarithm should always be greater than 0 and not equal to 1.

Most middle and high school logarithmic problems can be tackled using one or more of the following formulas:

\( \text{1. Definition: } \log_b(x) = y \iff b^y = x \\ \text{2. Logarithm of 1: } \log_b(1) = 0, \quad \text{for any } b > 0, \, b \neq 1 \\ \text{3. Logarithm of the base: } \log_b(b) = 1, \quad \text{for any } b > 0, \, b \neq 1 \\ \text{4. Logarithm of a power of the base: } \log_b(b^x) = x \\ \text{5. Base raised to a logarithm: } b^{\log_b(x)} = x \\ \text{6. Product Rule: } \log_b(xy) = \log_b(x) + \log_b(y) \\ \text{7. Quotient Rule: } \log_b\left(\frac{x}{y}\right) = \log_b(x) – \log_b(y) \\ \text{8. Power Rule: } \log_b(x^k) = k \cdot \log_b(x) \\ \text{9. Change of Base:} \log_b(x) = \frac{\log_k(x)}{\log_k(b)}, \quad \text{where } k > 0 \text{ and } k \neq 1 \)

Be sure to watch the entire video to fully understand the concepts, and feel free to let me know if you spot any inaccuracies. To test your understanding, there is also a quiz based on logarithms. The link to the quiz can be found below.

Links to the related quiz are here:

  1. Quiz 1

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