🌟 Level 4 - Topic 2: Advanced Exponential and Logarithmic Functions 🚀

1) Introduction to Advanced Exponential & Logarithmic Concepts 🤔

By now, you’ve learned the basics of exponential and logarithmic functions:

  • Exponential: \( f(x) = a \cdot b^x \)
  • Logarithmic: \( g(x) = \log_b x \)

At this advanced level, we dive deeper into:

  • More complex transformations and properties
  • Advanced equations involving exponentials and logarithms
  • Real-world applications (finance, decay/growth, advanced math problems)
  • Using logarithms to solve various forms of exponential equations

2) Transformations of Exponential & Logarithmic Functions ✨

2.1 Exponential Functions

A general exponential function can be written as:

\( f(x) = a \cdot b^{(kx + d)} + c \)

  • \( a \) controls vertical stretch and flipping (if negative).
  • \( b \) is the base (>1 for growth, 0<b<1 for decay).
  • \( k \) causes horizontal compression or stretch.
  • \( d \) causes a horizontal shift.
  • \( c \) shifts the function vertically.

Example 1:

\( f(x) = 3 \cdot 2^{(x-2)} - 1 \)

  • Base \( 2 \) → growth
  • Shift right by 2
  • Vertical shift down by 1
  • Stretch factor 3 → grows faster

2.2 Logarithmic Functions

A general logarithmic function can be written as:

\( g(x) = a \cdot \log_{b}(kx + d) + c \)

  • \( a \) is a vertical stretch/factor.
  • \( b \) is the logarithm base.
  • \( k \) and \( d \) shift/compress horizontally.
  • \( c \) shifts the function vertically.

Example 2:

\( g(x) = 2 \cdot \log_{3}(x - 1) + 3 \)

  • Log base 3
  • Shift right by 1
  • Vertical stretch of 2
  • Shift up by 3

3) Solving Advanced Exponential Equations 🧩

3.1 Using Logarithms

When exponents involve variables in complicated ways, we typically take logarithms on both sides.

Example 1: Solve

\( 2^{x+1} = 5^x \)

  1. Take the natural log of both sides:

    \( \ln(2^{x+1}) = \ln(5^x) \)

  2. Apply the log power rule:

    \( (x+1)\ln 2 = x \ln 5 \)

  3. Solve for \( x \):

    \( x \ln 2 + \ln 2 = x \ln 5 \)

    \( x(\ln 2 - \ln 5) = -\ln 2 \)

    \( x = \frac{-\ln 2}{\ln 2 - \ln 5} \)

3.2 Changing the Base or Using Log Properties

Sometimes rewriting bases simplifies the equation. For example, if \( 2^x = 8^x \) note that \( 8 = 2^3 \). Similarly, if \( e^x = 7^{2x-1} \), you can take logarithms or rewrite the base.

Example 2:

\( (3^x)^2 = 9^x \)

Since \( (3^x)^2 = 3^{2x} \) and \( 9 = 3^2 \) (so \( 9^x = 3^{2x} \)), both sides are equal, yielding infinite solutions if the domain is valid.


4) Advanced Logarithmic Equations 🤓

4.1 Combining Multiple Logs

You may encounter equations like:

\( \log_2(x-1) + \log_2(x+2) = 3 \)

  1. Combine the logs:

    \( \log_2[(x-1)(x+2)] = 3 \)

  2. Convert to exponential form:

    \( (x-1)(x+2) = 2^3 = 8 \)

  3. Solve the resulting quadratic (and check domain constraints).

Example 1: Solve

\( \log_3 (2x+1) - \log_3 (x-2) = 2 \)

  1. Combine the logs:

    \( \log_3 \Big(\frac{2x+1}{x-2}\Big)=2 \)

  2. Convert to exponential form:

    \( \frac{2x+1}{x-2} = 3^2 = 9 \)

  3. Solve \( 2x+1 = 9(x-2) \).

4.2 Log of Log

Sometimes we get nested logs:

\( \log_2(\log_3 x) = 1 \)

  1. Convert the outer log:

    \( \log_3 x = 2^1 = 2 \)

  2. Solve \( \log_3 x = 2 \) to get:

    \( x = 3^2 = 9 \)


5) Real-Life Applications 🌍

  • Compound Interest: \( A = P \Big(1+\frac{r}{n}\Big)^{nt} \) — use logarithms to solve for \( t \) or \( r \).
  • Carbon Dating (Exponential Decay): \( N(t) = N_0 e^{-kt} \) — take \(\ln\) to find time or rate.
  • pH Scale: \( pH = -\log_{10}[H^+] \).
  • Half-Life: Use logs to determine time in radioactive processes (e.g., find \( t \) when \( \frac{N}{N_0} = 0.5 \)).

Example: Given a half-life of 30 days, how long until only 10% remains? Set up and solve:

\( N(t)= N_0 \Big(\frac{1}{2}\Big)^{\frac{t}{30}} = 0.10 \, N_0 \)


6) Examples 🍀

  1. Exponential Growth: \( P(t)= 500 \cdot 1.08^t \). To find \( t \) such that \( P(t)=1000 \):

    \( 500 \cdot 1.08^t = 1000 \) → \( 1.08^t = 2 \) → \( t = \frac{\ln 2}{\ln 1.08} \)

  2. Logarithmic Equation: Solve

    \( \log(x) + \log(x-4) = 2 \)

    by combining logs to get \( \log[x(x-4)] = 2 \) then solving \( x(x-4)=10^2=100 \).
  3. Mixed Equation: Solve

    \( e^{2x-1} = 7^{x+3} \)

    using logarithms or rewriting the bases.

7) Practice Questions 🎯

7.1 Fundamental – Build Skills (10+)

  1. Transformation: Write the transformations for \( f(x)=3(2^{x-1})+2 \). Identify shifts and stretches.
  2. Rewrite: Express \( 16^x \) in terms of base 2, then simplify.
  3. Solve: \( 2^{x+2} = 5^{x} \) in terms of logarithms.
  4. Log Combine: Solve

    \( \log_4(x+3) + \log_4(x-1) = 2 \)

    and check domain constraints.
  5. Change Base: Rewrite \( \log_2 30 \) in terms of \(\ln\) or \(\log_{10}\).
  6. Vertical Shift: Sketch \( f(x)=2^{x}+3 \). Identify the horizontal asymptote and y-intercept.
  7. Exponential Equation: Solve

    \( 3^{2x-1} = 27^{x} \)

    for \( x \).
  8. Log Equation: Solve

    \( \ln(2x-1) = 4 \)

    for \( x \) (hint: exponentiate both sides).
  9. Applications: Given a population that doubles in 5 years with \( P(t)=100(2^{t/5}) \), find how long it takes to triple.
  10. pH Problem: Given

    \( pH = -\log_{10}[H^+] \)

    and \( pH=2.5 \), solve for \([H^+]\).

7.2 Challenging – Push Limits (5+)

  1. 🔥 Nested: Solve

    \( \log_2 (\log_3 (x+1)) = 2 \)

    step by step.
  2. 🔥 Compound Interest: Given

    \( A = 500 \Big(1+\frac{0.04}{4}\Big)^{4t} \)

    and \( A=1000 \), solve for \( t \) (round to 2 decimals).
  3. 🔥 Base Switch: Solve

    \( 5^{x+1} = (25)^{2x-3} \)

    for \( x \).
  4. 🔥 Exponential Intersection: Find the intersection(s) of:

    \( y=2^{x}+1 \) and \( y=3^{x-1}+2 \)

    (this may require logarithms or numerical approximation).
  5. 🔥 Decreasing Function: For a drug decaying as

    \( D(t)=100 \, e^{-0.2t} \)

    determine the time when the dose dips below 10 mg.

8) Summary

  • Transformations of exponentials and logarithms involve shifts, stretches, and compressions.
  • Advanced equations often require combining logs or applying logarithmic properties to solve exponential forms.
  • Real-world applications include finance, half-life calculations, pH scales, and more.

Master these advanced concepts to tackle higher-level math and physics with confidence! 🎉

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