- To expand log 2 (3x) log 2 (3) log 2 (x)
- To expand log 2 (12/x) log 2 (12) log 2 (x)
Is it possible to simplify logarithms?
Yes, as in algebraic equations, it is possible to simplify logarithms, but in different
ways. The following lists some examples:
- To simplify log 3 (x) log 3 (y) log 3 (xy)
- To simplify log 3 (6) log 3 (4) log 3 (6/4)
- To simplify 2log 3 (x) log 3 (x^2 )
How can the base of logarithms be changed?
The base of logarithms can be changed from one that is not 10 or eto an equivalent
logarithm with base 10 or e. The following gives the formula for such a transition, in
which a, b, and xare real positive numbers (but neither anor bare equal to 1, and xis
greater than 0):
Convert logaxto the base bby using the formula
(logbx)/(logba)What is an example of a log table?
An example of a log table can be seen in Appendix 2 at the back of this book.
How are equationswith exponents and logarithmssolved?
The way to solve an exponential equation is relatively easy: Take the log of both sides
of the equation, then solve for the variable. For example, to solve for xin the equation
ex60;
- First, take the natural log (ln) of both sides: ln(ex) ln(60)
- Simplify using the logarithmic rule #3 (see above) for the left side: x ln(e)
ln(60) - Then simplify again, since ln(e) 1 to: xln(60) 4.094344562
- And finally, check your answer (using log tables or your calculator) in the origi-
nal equation ex60: e4.09434456260 is definitely true.
The way to solve a logarithmic equation is equally easy: Just rewrite the equation
in exponential form and solve for the variable. For example, to solve for xin the equa-
tion ln(x) 11: - First, change both sides so they are exponents of the base e: eln(x)e^11
- When the bases of the exponent and logarithm are the same, the left part of the
equation becomes x, thus, it can be written: xe^11149