Introduces logarithms as the inverse of exponential functions. This stack focuses on converting between exponential and logarithmic forms, special logarithms (common and natural), and the basic features of a logarithmic graph.
How are logarithmic functions related to exponential functions?
A logarithmic function is the inverse of an exponential function. The logarithm of a number is the exponent to which another fixed value, the base, must be raised to produce that number.
How do you convert the exponential equation \(b^y = x\) into its equivalent logarithmic form?
The exponential equation \(b^y = x\) is equivalent to the logarithmic equation \(\log_b(x) = y\). Remember: the base of the log is the base of the power, and the log equals the exponent.
How do you convert the logarithmic equation \(\log_b(x) = y\) into its equivalent exponential form?
The logarithmic equation \(\log_b(x) = y\) is equivalent to the exponential equation \(b^y = x\).
Convert the exponential equation \(5^3 = 125\) to logarithmic form.
$$ \log_5(125) = 3 $$
Convert the logarithmic equation \(\log_7(49) = 2\) to exponential form.
$$ 7^2 = 49 $$
What is the "common logarithm" and what is its base?
The common logarithm has a base of \(10\). It is written as \(\log(x)\), where the base \(10\) is implied.
What is the "natural logarithm" and what is its base?
The natural logarithm has the natural base \(e\). It is written as \(\ln(x)\).
What is the x-intercept of the graph of \(f(x) = \log_b(x)\)?
The x-intercept is at the point \((1, 0)\), because \(\log_b(1) = 0\) for any valid base \(b\).
What is the vertical asymptote of the graph of \(f(x) = \log_b(x)\) (assuming no horizontal shift)?
The vertical asymptote is the y-axis, which is the line \(x = 0\).
What is the domain of a standard logarithmic function \(f(x) = \log_b(x)\)?
The domain is all positive real numbers, \((0, \infty)\), because you cannot take the logarithm of a non-positive number.
What is the range of a standard logarithmic function \(f(x) = \log_b(x)\)?
The range is all real numbers, \((-\infty, \infty)\).
Evaluate \(\log(1000)\).
Since \(\log(1000)\) means \(\log_{10}(1000)\), we are asking \(10^? = 1000\). Since \(10^3 = 1000\), the answer is \(3\).
What is the approximate value of \(\ln(1)\)?
\(\ln(1) = 0\), because \(e^0 = 1\). This is true for any logarithm base: \(\log_b(1) = 0\).
How is the graph of \(y = \log_b(x)\) geometrically related to the graph of its inverse, \(y = b^x\)?
The graph of \(y = \log_b(x)\) is a reflection of the graph of \(y = b^x\) across the line \(y = x\).
