IIT Madras
Mathematics 1
Logarithmic Functions Applications

Logarithmic Functions Applications

A well-defined collection of distinct objects called elements or members.

https://youtu.be/XSJ1QZ0MHOs

Learning Outcomes:

  1. To learn the applications of laws of logarithm

Exercise Questions

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Good morning! Here in India on this Monday, let’s explore these questions. They are great examples of how logarithms are used to solve real-world problems involving very large numbers and complex interest calculations.

Core Concepts: Logarithms in Application

  1. Counting Digits in a Large Number ($N$): Logarithms are the perfect tool for this. The number of digits in any positive integer $N$ is given by the formula:

    $$\text{Number of digits} = \lfloor \log_{10}(N) \rfloor + 1$$

    Where $\lfloor x \rfloor$ is the “floor function,” which means you take only the integer part of the number (e.g., $\lfloor 57.6 \rfloor = 57$). To use this, we often need the Power Rule of logarithms: $\log(b^p) = p \cdot \log(b)$.

  2. Compound Interest Formula: The formula for an investment that is compounded multiple times per year is:

    $$A = P \left(1 + \frac{r}{k}\right)^{kt}$$

    where:

    • $A$ = the final amount
    • $P$ = the principal (initial amount)
    • $r$ = the annual interest rate (as a decimal)
    • $k$ = the number of times interest is compounded per year
    • $t$ = the number of years

Question 1: Number of Digits

The Question: How many digits are there in $15^{77}$?

Detailed Solution:

Note: There appears to be a significant typo in this question. The number $15^{77}$ is enormous and has 91 digits. The provided options and the accepted answer of “58” suggest the question was intended to be for a different power, most likely $15^{49}$. I will solve the question as it was likely intended.

Solving the Corrected Question: “How many digits are there in $15^{49}$?”

  1. Apply the formula: The number of digits is $\lfloor \log_{10}(15^{49}) \rfloor + 1$.

  2. Use the Power Rule of logarithms:

    • $\log_{10}(15^{49}) = 49 \times \log_{10}(15)$

  3. Calculate the value of $\log_{10}(15)$:

    • We know $\log_{10}(15) = \log_{10}(3 \times 5) = \log_{10}(3) + \log_{10}(5)$.
    • Using standard log values: $\log_{10}(3) \approx 0.4771$ and $\log_{10}(5) \approx 0.6990$.
    • $\log_{10}(15) \approx 0.4771 + 0.6990 = 1.1761$.
    • (A more precise value is $\log_{10}(15) \approx 1.17609$)

  4. Multiply by the exponent:

    • $49 \times 1.17609 \approx 57.628$

  5. Find the number of digits:

    • Number of digits = $\lfloor 57.628 \rfloor + 1$
    • Number of digits = $57 + 1 = 58$.

Final Answer: There are 58 digits in the number $15^{49}$.

Question 2: Compound Interest

The Question: Suppose a certain amount of money $M$ is invested in a mutual fund at an annual rate of interest of 5%. How long(approximately) does it take to triple the initial investment, assuming interest is compounded thrice a year?

Detailed Solution:

  1. Identify the variables for the compound interest formula, $A = P \left(1 + \frac{r}{k}\right)^{kt}$:

    • Principal amount, $P = M$.
    • Final amount (triple the initial), $A = 3M$.
    • Annual rate, $r = 5% = 0.05$.
    • Compounding frequency, $k = 3$ (thrice a year).
    • Time in years, $t$, is what we need to find.

  2. Set up the equation:

    $$3M = M \left(1 + \frac{0.05}{3}\right)^{3t}$$

  3. Simplify the equation:

    • Divide both sides by $M$: $$3 = \left(1 + \frac{0.05}{3}\right)^{3t}$$
    • Calculate the value inside the parenthesis: $$3 = (1 + 0.01666...)^{3t} \approx (1.01667)^{3t}$$

  4. Use logarithms to solve for the exponent, $t$:

    • Take the natural logarithm ($\ln$) of both sides: $$\ln(3) = \ln((1.01667)^{3t})$$
    • Apply the Power Rule of logarithms to bring the exponent down: $$\ln(3) = 3t \cdot \ln(1.01667)$$

  5. Isolate and calculate $t$:

    $$t = \frac{\ln(3)}{3 \cdot \ln(1.01667)}$$
    • Using a calculator for the log values:
      • $\ln(3) \approx 1.0986$
      • $\ln(1.01667) \approx 0.01653$
    • $$t \approx \frac{1.0986}{3 \times 0.01653} = \frac{1.0986}{0.04959} \approx 22.15 \text{ years}$$

Final Answer: It will take approximately 22 years to triple the initial investment.

Logarithmic Functions - GraphsLogarithmic Functions Properties - 2