Let
$\text{\hspace{0.17em}}m={\mathrm{log}}_{b}M\text{\hspace{0.17em}}$ and
$\text{\hspace{0.17em}}n={\mathrm{log}}_{b}N.\text{\hspace{0.17em}}$ In exponential form, these equations are
$\text{\hspace{0.17em}}{b}^{m}=M\text{\hspace{0.17em}}$ and
$\text{\hspace{0.17em}}{b}^{n}=N.\text{\hspace{0.17em}}$ It follows that
Note that repeated applications of the product rule for logarithms allow us to simplify the logarithm of the product of any number of factors. For example, consider
$\text{\hspace{0.17em}}{\mathrm{log}}_{b}(wxyz).\text{\hspace{0.17em}}$ Using the product rule for logarithms, we can rewrite this logarithm of a product as the sum of logarithms of its factors:
For quotients, we have a similar rule for logarithms. Recall that we use the
quotient rule of exponents to combine the quotient of exponents by subtracting:
$\text{\hspace{0.17em}}{x}^{\frac{a}{b}}={x}^{a-b}.\text{\hspace{0.17em}}$ The
quotient rule for logarithms says that the logarithm of a quotient is equal to a difference of logarithms. Just as with the product rule, we can use the inverse property to derive the quotient rule.
Given any real number
$\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ and positive real numbers
$\text{\hspace{0.17em}}M,$$N,$ and
$\text{\hspace{0.17em}}b,$ where
$\text{\hspace{0.17em}}b\ne 1,$ we will show
Let
$\text{\hspace{0.17em}}m={\mathrm{log}}_{b}M\text{\hspace{0.17em}}$ and
$\text{\hspace{0.17em}}n={\mathrm{log}}_{b}N.\text{\hspace{0.17em}}$ In exponential form, these equations are
$\text{\hspace{0.17em}}{b}^{m}=M\text{\hspace{0.17em}}$ and
$\text{\hspace{0.17em}}{b}^{n}=N.\text{\hspace{0.17em}}$ It follows that
For example, to expand
$\text{\hspace{0.17em}}\mathrm{log}\left(\frac{2{x}^{2}+6x}{3x+9}\right),$ we must first express the quotient in lowest terms. Factoring and canceling we get,
I am a carpenter and I have to cut and assemble a conventional roof line for a new home. The dimensions are: width 30'6" length 40'6". I want a 6 and 12 pitch. The roof is a full hip construction. Give me the L,W and height of rafters for the hip, hip jacks also the length of common jacks.
like Deadra, show me the step by step order of operation to alive for b
John
A laser rangefinder is locked on a comet approaching Earth. The distance g(x), in kilometers, of the comet after x days, for x in the interval 0 to 30 days, is given by g(x)=250,000csc(π30x). Graph g(x) on the interval [0, 35]. Evaluate g(5) and interpret the information. What is the minimum distance between the comet and Earth? When does this occur? To which constant in the equation does this correspond? Find and discuss the meaning of any vertical asymptotes.