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
\( 2^{x+1} = 5^x \)
- Take the natural log of both sides:
\( \ln(2^{x+1}) = \ln(5^x) \)
- Apply the log power rule:
\( (x+1)\ln 2 = x \ln 5 \)
- 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:
\( \log_2(x-1) + \log_2(x+2) = 3 \)
- Combine the logs:
\( \log_2[(x-1)(x+2)] = 3 \)
- Convert to exponential form:
\( (x-1)(x+2) = 2^3 = 8 \)
- Solve the resulting quadratic (and check domain constraints).
Example 1: Solve
\( \log_3 (2x+1) - \log_3 (x-2) = 2 \)
- Combine the logs:
\( \log_3 \Big(\frac{2x+1}{x-2}\Big)=2 \)
- Convert to exponential form:
\( \frac{2x+1}{x-2} = 3^2 = 9 \)
- Solve \( 2x+1 = 9(x-2) \).
4.2 Log of Log
Sometimes we get nested logs:
\( \log_2(\log_3 x) = 1 \)
- Convert the outer log:
\( \log_3 x = 2^1 = 2 \)
- 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 🍀
- Exponential Growth: \( P(t)= 500 \cdot 1.08^t \). To find \( t \) such that \( P(t)=1000 \):
\( 500 \cdot 1.08^t = 1000 \) → \( 1.08^t = 2 \) → \( t = \frac{\ln 2}{\ln 1.08} \)
- Logarithmic Equation: Solve
\( \log(x) + \log(x-4) = 2 \)
by combining logs to get \( \log[x(x-4)] = 2 \) then solving \( x(x-4)=10^2=100 \). - Mixed Equation: Solve
\( e^{2x-1} = 7^{x+3} \)
using logarithms or rewriting the bases.
7) Practice Questions 🎯
7.1 Fundamental – Build Skills (10+)
- Transformation: Write the transformations for \( f(x)=3(2^{x-1})+2 \). Identify shifts and stretches.
- Rewrite: Express \( 16^x \) in terms of base 2, then simplify.
- Solve: \( 2^{x+2} = 5^{x} \) in terms of logarithms.
- Log Combine: Solve
\( \log_4(x+3) + \log_4(x-1) = 2 \)
and check domain constraints. - Change Base: Rewrite \( \log_2 30 \) in terms of \(\ln\) or \(\log_{10}\).
- Vertical Shift: Sketch \( f(x)=2^{x}+3 \). Identify the horizontal asymptote and y-intercept.
- Exponential Equation: Solve
\( 3^{2x-1} = 27^{x} \)
for \( x \). - Log Equation: Solve
\( \ln(2x-1) = 4 \)
for \( x \) (hint: exponentiate both sides). - Applications: Given a population that doubles in 5 years with \( P(t)=100(2^{t/5}) \), find how long it takes to triple.
- pH Problem: Given
\( pH = -\log_{10}[H^+] \)
and \( pH=2.5 \), solve for \([H^+]\).
7.2 Challenging – Push Limits (5+)
- 🔥 Nested: Solve
\( \log_2 (\log_3 (x+1)) = 2 \)
step by step. - 🔥 Compound Interest: Given
\( A = 500 \Big(1+\frac{0.04}{4}\Big)^{4t} \)
and \( A=1000 \), solve for \( t \) (round to 2 decimals). - 🔥 Base Switch: Solve
\( 5^{x+1} = (25)^{2x-3} \)
for \( x \). - 🔥 Exponential Intersection: Find the intersection(s) of:
\( y=2^{x}+1 \) and \( y=3^{x-1}+2 \)
(this may require logarithms or numerical approximation). - 🔥 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! 🎉