Introduces the basic definition of a logarithm as the inverse of an exponential function. Covers converting between logarithmic and exponential forms, and defines common and natural logarithms.
What is the fundamental relationship between the logarithmic equation \(y = \log_b(x)\) and an exponential equation?
They are inverse forms. The equation \(y = \log_b(x)\) is equivalent to \(b^y = x\). The base \(b\) must be positive and not equal to \(1\) (\(b > 0\), \(b \neq 1\)). The argument \(x\) must be positive (\(x>0\)).
Convert the exponential equation \(5^3 = 125\) to its logarithmic form.
$$\log_5(125) = 3$$
Convert the logarithmic equation \(\log_2(16) = 4\) to its exponential form.
$$2^4 = 16$$
What is a 'common logarithm' and what is its base?
A common logarithm is a logarithm with base \(10\). It is usually written without the base, as \(\log(x)\), which is shorthand for \(\log_{10}(x)\).
What is a 'natural logarithm' and what is its base?
A natural logarithm is a logarithm with base \(e\) (Euler's number). It is written as \(\ln(x)\), which is shorthand for \(\log_e(x)\). The value of \(e\) is approximately \(2.718\).
Evaluate \(\log_7(49)\).
The answer is \(2\), because \(7^2 = 49\).
Evaluate \(\log(1000)\).
This is a common log (base \(10\)). The answer is \(3\), because \(10^3 = 1000\).
What is the domain of the function \(f(x) = \log_b(x)\)?
The domain is all positive real numbers, \((0, \infty)\). You cannot take the logarithm of a non-positive number (zero or a negative number).
