Condensing and Expanding Logarithmic Expressions

Front

What is the property of logarithms that allows you to condense the expression \(2\log_3u - \log_3z\)?

Back

The property states that \(a\log_b(x) = \log_b(x^a)\) and \(\log_b(x) - \log_b(y) = \log_b\left(\frac{x}{y}\right)\). Thus, \(2\log_3u - \log_3z = \log_3\left(\frac{u^2}{z}\right)\).

Front

Convert the logarithmic equation \(\log_x25=3\) into its exponential form.

Back

The exponential form is \(x^3=25\).

Front

How do you expand the logarithmic expression \(\log\left(\frac{x}{y}\right)\)?

Back

Using the property \(\log\left(\frac{a}{b}\right) = \log a - \log b\), the expansion is \(\log x - \log y\).

Front

What is the condensed form of the expression \(\log_5 7 + 2\log_5 x\)?

Back

Using the properties of logarithms, the condensed form is \(\log_5(7x^2)\).

Front

How do you condense the expression \(2\log_6a + \log_6b - 3\log c\)?

Back

The condensed form is \(\log_6\left(\frac{a^2b}{c^3}\right)\).

Front

What is the definition of a logarithm?

Back

A logarithm is the exponent to which a base must be raised to produce a given number. For example, if \(b^y = x\), then \(\log_b x = y\).

Front

What is the change of base formula for logarithms?

Back

The change of base formula states that \(\log_b a = \frac{\log_k a}{\log_k b}\) for any positive base \(k\).

Front

What is the product property of logarithms?

Back

The product property states that \(\log_b(xy) = \log_b x + \log_b y\).

Front

What is the quotient property of logarithms?

Back

The quotient property states that \(\log_b\left(\frac{x}{y}\right) = \log_b x - \log_b y\).

Front

What is the power property of logarithms?

Back

The power property states that \(\log_b(x^k) = k\log_b x\).