Combine the following logarithmic equations into one:

Combine the three logs together:

Use exponent rule: log 4xy^{3} Simplify: log 64x^{3}y^{3}

log 64x^{3}y^{3}

Example 3

Simplify the following exponential function to logarithmic form, then simplify irrationals to three decimal places and convert back to exponential form:

y = 10^{2x}7^{x}

We can take either the base-10 or base-7 log. Let's take 10, because we like base-10:

log y = log 10^{2x}7^{x}

Separate the logarithms using the sum of logs rule:

log y = log 10^{2x} + log 7^{x}

The first term on the right cancels:

log y = 2x + log 7^{x}

Pull out the exponent from the second term using the exponent in log rule:

log y = 2x + x log 7

Simplify and add x's: log y = 2.846x

Exponentiate both sides with bases of 10: 10^{log y} = 10^{2.846x}

Left-hand side cancels: y = 10^{2.846x} y = 10^{2.846x}

Example 4

Is y = log_{2} 4x equivalent to 2^{y – 2} = x

If so, show how.

Separate the logs on the right side: y = log_{2}4 + log_{2}x Solve the left-hand log: y = 2 + log_{2}x Isolate the term with x: y – 2 = log_{2}x Exponentiate both sides with a base of 2:

2^{y – 2} = 2^{log2x}

Right-hand side exponent and log cancel: 2^{y – 2} = x

Yes.

Example 5

Expand the following logarithmic equation:

First separate the numerator and denominator: log 2x^{2} – log 4x Then remove the exponent from the first log: 2 log 2x – log 4x

2 log 2x – log 4x

Example 6

Expand the following equation: log 27x^{2}

Separate two terms: log 27 + log x^{2} Rewrite in equivalent form: log 3^{3} + log x^{2} Pull out exponents: 3 log 3 + 2 log x

3 log 3 + 2 log x

Example 7

Expand the following log equation, and write it using only base-10 logarithmic functions. Solve irrationals to 3 decimal places:

y = ln (4^{e})(3x + 1)

First, separate into two terms using the sum of logs rule:

y = ln 4^{e} + ln(3x+1)

Use the change of base rule on both terms:

Solve irrationals:

Example 8

Is equivalent to 10^{y} = 81x^{2}?

If so, show how.

First, multiply by two on both sides: y = 2 log 9x Use the property of exponents in logs: y = log 9x^{2} Distribute the exponent: y = log 81x^{2}

Exponentiate both sides with a base of 10: 10^{y} = 81x^{2} Yes.

Example 9

Can the following expression be simplified: y = (log x)(log 2)?

Because there are no logarithmic laws that can combine products of logs, and because there is only one term in each log that cannot be manipulated in any meaningful way, the expression cannot be manipulated.