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Shortcut Techniques: Functions Video Lecture | Quantitative for GMAT
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FAQs on Shortcut Techniques: Functions Video Lecture - Quantitative for GMAT
| 1. What are the basic properties of logarithms that can help in simplifying logarithmic expressions? |  |
| 2. How can I quickly solve logarithmic equations using these properties? | |
Ans. To quickly solve logarithmic equations, first, use the properties of logarithms to combine or expand terms as needed. Then, convert the logarithmic equation into its exponential form. For example, if you have $\log_b x = y$, you can rewrite it as $x = b^y$. This approach helps isolate the variable and solve for it more easily.
| 3. What are some common logarithmic identities I should memorize for exams? | |
Ans. Some common logarithmic identities to memorize include:
1. $\log_b b = 1$: The logarithm of a base to itself equals 1.
2. $\log_b 1 = 0$: The logarithm of 1 in any base equals 0.
3. $\log_b (b^n) = n$: The logarithm of a base raised to a power equals that power.
4. $\log_b (b^{1/n}) = \frac{1}{n}$: The logarithm of a base to the fractional power equals the reciprocal of that fraction.
Familiarity with these identities can significantly speed up solving logarithmic problems.
| 4. What techniques can I use to graph logarithmic functions effectively? | |
Ans. When graphing logarithmic functions, consider the following techniques:
1. Identify the vertical asymptote, which occurs at $x = 0$ for all logarithmic functions.
2. Plot key points, such as $(1, 0)$ since $\log_b 1 = 0$, and $(b, 1)$ since $\log_b b = 1$.
3. Understand the general shape of logarithmic graphs, which increase slowly and are continuous for $x > 0$.
4. Use transformations to shift or stretch the graph based on changes in the function's equation.
| 5. How do logarithms apply in real-world situations, such as in finance or science? | |
Ans. Logarithms have various real-world applications, including:
1. In finance, logarithms are used to calculate compound interest and to analyze exponential growth in investments.
2. In science, logarithmic scales are often used, such as the Richter scale for earthquake intensity and the pH scale for acidity.
3. Logarithmic functions help model phenomena that grow or decay at rates proportional to their current value, such as population growth or radioactive decay.
Understanding these applications can provide practical insights into how logarithms are utilized in different fields.
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