Introduction to Logarithms
In your previous studies, you learned how to use exponents to grow numbers—for example, knowing that \(2^3 = 8\). But what if we need to go backward? What if we know the result is \(8\) and the base is \(2\), but we need to find the exponent? This is where logarithms come in. Logarithms are simply the "inverse" of exponents. They help us solve for the unknown power in an equation.
Think of it this way: Exponents ask, "What do I get when I multiply this base by itself this many times?" Logarithms ask, "How many times must I multiply this base by itself to get this specific result?"
The Relationship Between Exponents and Logarithms
Every exponential statement can be rewritten as a logarithmic statement. Understanding this "switch" is the most important skill in this chapter.
If \(a^x = b\), then \(\log_a b = x\).
- \(a\) is the base (it stays the base in both forms).
- \(x\) is the exponent (the answer to a log is always an exponent).
- \(b\) is the argument or the result.
Example: Since \(10^2 = 100\), we can write \(\log_{10} 100 = 2\).
Example: Since \(2^5 = 32\), we can write \(\log_2 32 = 5\).
Quick Tip: The "Loop" Method
To convert from \(\log_a b = x\) back to exponential form, imagine a loop starting at the base \(a\), going to the answer \(x\), and ending at the argument \(b\). This gives you \(a^x = b\).
Two Special Bases: Base 10 and Base \(e\)
While a logarithm can have almost any positive number as a base (except \(1\)), there are two bases you will use most often in the IB DP curriculum:
- The Common Logarithm (Base 10): Written as \(\log_{10} x\) or often just \(\log x\). If you see a log without a base, assume it is base \(10\).
- The Natural Logarithm (Base \(e\)): Written as \(\ln x\). This uses the mathematical constant \(e \approx 2.718\). It is extremely important in higher-level calculus and modeling.
Did you know? Your calculator has specific buttons for both \(\log\) and \(\ln\). For SL and HL students, being able to evaluate these using technology is a required skill (Syllabus SL 1.5).
The Laws of Logarithms
Just as there are "Laws of Exponents" (which you can review in the Standard form and laws of exponents chapter), there are Laws of Logarithms. These rules allow us to combine or break apart log expressions to make them easier to solve.
1. The Product Law: \(\log_a (xy) = \log_a x + \log_a y\)
(Adding logs with the same base is like multiplying their arguments.)
2. The Quotient Law: \(\log_a \left(\frac{x}{y}\right) = \log_a x - \log_a y\)
(Subtracting logs with the same base is like dividing their arguments.)
3. The Power Law: \(\log_a (x^k) = k \log_a x\)
(This is the "secret weapon" for solving equations! It allows you to move an exponent down to the front as a multiplier.)
Important Basic Identities:
- \(\log_a a = 1\) (Because \(a^1 = a\))
- \(\log_a 1 = 0\) (Because \(a^0 = 1\))
- \(\log_a (a^x) = x\)
- \(a^{\log_a x} = x\)
The Change of Base Formula
Sometimes you might encounter a log with a base your calculator doesn't have a specific button for (like \(\log_3 7\)). You can change it to any base you like (usually base \(10\) or \(e\)) using this formula:
\(\log_a x = \frac{\log_b x}{\log_b a}\)
Example: To calculate \(\log_3 7\), you can type \(\frac{\ln 7}{\ln 3}\) or \(\frac{\log 7}{\log 3}\) into your calculator. Both will give you the same result!
Solving Exponential Equations
This is where everything comes together. An exponential equation is one where the unknown variable is in the exponent, like \(5^x = 100\).
Method 1: Using Logarithms (Analytic Approach)
Don't worry if the numbers don't look like perfect squares or cubes! Here is the step-by-step process:
- Isolate the exponential part of the equation.
- Take the log (usually \(\ln\) or \(\log_{10}\)) of both sides.
- Use the Power Law to bring the \(x\) down in front.
- Solve for \(x\) using basic algebra.
Example: Solve \(3^x = 20\).
\(\ln(3^x) = \ln(20)\)
\(x \ln 3 = \ln 20\)
\(x = \frac{\ln 20}{\ln 3} \approx 2.73\)
Method 2: Equations with "Hidden" Quadratics
Sometimes you will see equations like \(e^{2x} - 5e^x + 6 = 0\). This looks scary, but it’s actually a quadratic in disguise! If you let \(u = e^x\), then the equation becomes \(u^2 - 5u + 6 = 0\).
1. Factor the quadratic: \((u - 2)(u - 3) = 0\).
2. Substitute back: \(e^x = 2\) or \(e^x = 3\).
3. Solve for \(x\): \(x = \ln 2\) or \(x = \ln 3\).
Method 3: Technology (GDC)
For Paper 2, you can solve complex equations graphically. Graph the left side as \(y_1\) and the right side as \(y_2\). The \(x\)-coordinate of the point of intersection is your solution.
Common Mistakes to Avoid
- Mistake: Thinking \(\log(x + y) = \log x + \log y\).
Correction: There is no rule for the log of a sum! The rule only works for the log of a product (\(\log xy\)). - Mistake: Trying to find the log of a negative number.
Correction: The argument of a logarithm must always be positive (\(b > 0\)). If you get a solution that makes the argument negative, it is an "extraneous" solution and must be rejected. - Mistake: Confusing \(\log(\frac{x}{y})\) with \(\frac{\log x}{\log y}\).
Correction: \(\log(\frac{x}{y}) = \log x - \log y\). The second expression is what we use in the Change of Base formula.
Key Takeaways
1. Definition: \(\log_a b = x\) is the same as \(a^x = b\).
2. Log Laws: Use them to condense expressions before solving.
3. Power Law: This is your primary tool for "dropping" an exponent down so you can solve for it.
4. Technology: Use your GDC to evaluate logs and find intersections of curves for complex equations.
5. Natural Log: Remember that \(\ln x\) is just a log with base \(e\); it follows all the same rules!