Welcome to Exponential and Logarithms (Unit P3)!
Hello! This chapter is incredibly important. Exponentials and logarithms are not just abstract mathematical concepts; they describe how things grow (like populations, investments, and diseases) or decay (like radioactive materials and drug concentrations in the bloodstream).
Don't worry if these functions seem strange at first. We are moving from base 10 (the standard logarithm you might have met before) to the natural base, \(e\). Once you understand the relationship between \(e^x\) and \(\ln x\), the rest is simply applying familiar rules in new ways.
Section 1: The Exponential Function – \(y = e^x\)
What is \(e\)? Euler's Number
In mathematics, there is a special number known as Euler's Number (or the natural base), denoted by the letter \(e\).
- It is an irrational number, approximately equal to \(e \approx 2.71828\).
- It naturally arises when calculating growth that is compounded continuously. Think of it as the ultimate rate of growth.
- The function \(f(x) = e^x\) is called the exponential function.
Understanding the Graph of \(y = e^x\)
The graph of \(y = e^x\) shows extremely fast growth.
- Y-intercept: When \(x=0\), \(y = e^0 = 1\). The graph passes through \((0, 1)\).
- Domain: All real numbers (\(x \in \mathbb{R}\)).
- Range: \(y > 0\). The graph is always positive.
- Asymptote: The graph approaches the x-axis (\(y=0\)) but never touches or crosses it. \(y=0\) is a horizontal asymptote.
Analogy: Imagine compound interest. \(y = e^x\) shows the value when the interest is calculated every single moment. This rapid growth is why the function curves upwards so steeply!
Key Takeaway for \(e^x\): It’s the standard exponential growth function, always positive, and crosses the y-axis at 1.
Section 2: The Natural Logarithm – \(y = \ln x\)
Logs as Inverses
The natural logarithm, written as \(\ln x\), is the inverse function of the exponential function \(e^x\).
What does inverse mean? If a function \(f\) takes 2 to 7, the inverse function \(f^{-1}\) takes 7 back to 2.
Since exponentials and natural logarithms are inverses, they undo each other:
1. If \(y = e^x\), then \(\ln y = x\).
2. If \(y = \ln x\), then \(e^y = x\).
Crucial Identity: \[\ln(e^x) = x \quad \text{and} \quad e^{\ln x} = x\]
Understanding the Graph of \(y = \ln x\)
Since \(y = \ln x\) is the inverse of \(y = e^x\), its graph is a reflection of \(y = e^x\) across the line \(y = x\).
- X-intercept: When \(y=0\), \(0 = \ln x\), which means \(e^0 = x\). So, \(x=1\). The graph passes through \((1, 0)\).
- Domain (Restriction!): \(x > 0\). You can only take the logarithm of a positive number.
- Range: All real numbers (\(y \in \mathbb{R}\)).
- Asymptote: The graph approaches the y-axis (\(x=0\)) but never touches or crosses it. \(x=0\) is a vertical asymptote.
Common Mistake to Avoid: Never try to calculate \(\ln(0)\) or \(\ln(\text{negative number})\). It is undefined! Always check the domain when solving logarithmic equations.
Key Takeaway for \(\ln x\): It tells you the power you need to raise \(e\) to in order to get \(x\). It is only defined for positive values of \(x\).
Section 3: Laws of Natural Logarithms
The rules (laws) for \(\ln x\) are exactly the same as the rules for logarithms in any other base (like \(\log_{10}\)). Make sure you have these memorized!
The Three Essential Laws
Let \(a\) and \(b\) be positive numbers.
1. The Product Law (Addition)
The log of a product is the sum of the logs.
\[\ln(ab) = \ln a + \ln b\]
Memory Aid: Multiply inside \(\rightarrow\) Add outside.
2. The Quotient Law (Subtraction)
The log of a quotient is the difference of the logs.
\[\ln\left(\frac{a}{b}\right) = \ln a - \ln b\]
Tip: The numerator's log comes first.
3. The Power Law (Multiplication)
The log of a number raised to a power moves the power to the front. This is the most frequently used law when solving equations!
\[\ln(a^k) = k \ln a\]
Quick Review of Log Values
- \(\ln(1) = 0\) (because \(e^0 = 1\))
- \(\ln(e) = 1\) (because \(e^1 = e\))
Key Takeaway on Laws: Use the laws to condense several log terms into a single log term, or to expand a complex log term into simpler parts.
Section 4: Solving Equations Involving \(e^x\) and \(\ln x\)
Solving these equations relies entirely on using the inverse relationship. We need to "undo" the function to solve for \(x\).
Type 1: Solving Exponential Equations (Involving \(e^x\))
To solve for \(x\) when it is in the exponent, we take the natural logarithm (\(\ln\)) of both sides.
Step-by-Step Example: Solve \(5e^{2x} - 3 = 12\)
- Isolate the exponential term: Get \(e^{\text{something}}\) by itself.
\[5e^{2x} = 15\] \[e^{2x} = 3\] - Take \(\ln\) of both sides: This brings the exponent down.
\[\ln(e^{2x}) = \ln(3)\] \[2x = \ln(3)\] - Solve for \(x\):
\[x = \frac{\ln(3)}{2}\] - Calculate (if asked for a decimal answer): \(x \approx 0.549\) (to 3 s.f.)
Encouragement: The key is always Step 1! If you can isolate the term, the rest is straightforward application of the identity \(\ln(e^{f(x)}) = f(x)\).
Type 2: Solving Logarithmic Equations (Involving \(\ln x\))
To solve for \(x\) when it is inside a logarithm, we use the inverse function \(e^{\text{something}}\) on both sides.
Step-by-Step Example: Solve \(\ln(x-1) + 4 = 6\)
- Isolate the logarithmic term: Get \(\ln(\text{something})\) by itself.
\[\ln(x-1) = 2\] - Use the inverse (Raise \(e\) to the power of both sides):
\[e^{\ln(x-1)} = e^2\] \[x-1 = e^2\] - Solve for \(x\):
\[x = e^2 + 1\] - Check your answer (Essential!): Since the domain of \(\ln\) is \(x>0\), we must check that the argument (\(x-1\)) is positive. Since \(e^2 \approx 7.39\), \(x \approx 8.39\). The argument \(x-1 = e^2\), which is positive. The solution is valid.
Important Note on Extraneous Solutions: If you had solved an equation and ended up with a solution like \(x = 0.5\), and the original equation involved \(\ln(x-1)\), then the argument would be \(0.5 - 1 = -0.5\). Since \(\ln(-0.5)\) is undefined, \(x=0.5\) would be an extraneous solution (a false solution) and must be rejected.
Key Takeaway on Solving: Isolate the term first. Use \(\ln\) to kill \(e\). Use \(e\) to kill \(\ln\). ALWAYS check the domain of your log solutions.
Section 5: Modelling Real-World Situations
One of the main applications of P3 exponentials is modelling scenarios like population growth, radioactive decay, and cooling/heating. These models usually take the form:
\[P = Ae^{kt}\]
Where:
- \(P\) is the quantity (population, temperature, mass, etc.)
- \(t\) is the time (usually in years, hours, or seconds)
- \(A\) is the initial quantity (when \(t=0\)). Since \(e^0 = 1\), if \(t=0\), \(P=A\).
- \(k\) is the rate constant.
- If \(k > 0\), it represents exponential growth.
- If \(k < 0\), it represents exponential decay.
Using Data to Find Constants (Linearisation)
Sometimes, you are given experimental data points (\(x, y\)) that follow a complex exponential relationship, and you need to find the constants \(A\) and \(k\).
To do this, we transform the relationship into a linear form (\(Y = mX + C\)). This is a vital P3 skill!
Case 1: Exponential Model \(y = Ae^{kx}\)
This model relates \(y\) and \(x\). We want to transform it into the linear form \(Y = mX + C\).
- Take the natural logarithm of both sides:
\[\ln y = \ln(Ae^{kx})\] - Apply the Product Law:
\[\ln y = \ln A + \ln(e^{kx})\] - Apply the Power Law (\(\ln(e^{kx}) = kx\)):
\[\ln y = \ln A + kx\] - Rearrange into \(Y = mX + C\) format:
\[\ln y = kx + \ln A\]
If you plot your data:
- The Y-axis variable is \(Y = \ln y\).
- The X-axis variable is \(X = x\).
- The gradient is \(m = k\).
- The Y-intercept is \(C = \ln A\).
This means: If you plot \(\ln y\) against \(x\) and get a straight line, your data fits the \(y = Ae^{kx}\) model! You can find \(k\) directly from the gradient, and \(A\) by calculating \(e^C\).
Did you know? This technique is used constantly in labs when analyzing chemical reactions or physical decay processes. It turns a complicated curve into a simple straight line, making calculations much easier!
Case 2: Power Law Model \(y = Ax^n\) (Common Trap!)
Although this doesn't involve \(e^x\), you must use \(\ln\) to linearise it.
- Take \(\ln\) of both sides:
\[\ln y = \ln(Ax^n)\] - Apply the Product Law:
\[\ln y = \ln A + \ln(x^n)\] - Apply the Power Law:
\[\ln y = n \ln x + \ln A\]
If you plot your data:
- The Y-axis variable is \(Y = \ln y\).
- The X-axis variable is \(X = \ln x\).
- The gradient is \(m = n\).
- The Y-intercept is \(C = \ln A\).
Be careful! Notice the difference between Case 1 (plot \(\ln y\) vs \(x\)) and Case 2 (plot \(\ln y\) vs \(\ln x\)). Read the exam question carefully to determine which variables you need to plot!
Key Takeaway on Modelling: Use the laws of logarithms to convert a complicated non-linear model into the linear form \(Y = mX + C\). This allows you to find unknown constants \(A\) and \(k\) (or \(A\) and \(n\)) using gradient and intercept calculations.
Chapter Summary Checklist
- I understand that \(e\) is the natural base and is approximately \(2.718\).
- I know the inverse relationship: \(\ln(e^x) = x\) and \(e^{\ln x} = x\).
- I can apply the Product, Quotient, and Power Laws to \(\ln\).
- I can solve equations involving \(e^x\) by isolating the term and using \(\ln\).
- I can solve equations involving \(\ln x\) by isolating the term and using \(e\), and I always check my solution domain.
- I can use \(\ln\) to linearise models of the form \(y = Ae^{kx}\) and \(y = Ax^n\).
Keep practicing those transformation and solving skills! You've got this!