The exponential function
$f\left(x\right)={b}^{x}$ is one-to-one, with domain
$(\text{\u2212}\infty ,\infty )$ and range
$\left(0,\infty \right).$ Therefore, it has an inverse function, called the
logarithmic function with base$b.$ For any
$b>0,b\ne 1,$ the logarithmic function with base
b , denoted
${\text{log}}_{b},$ has domain
$(0,\infty )$ and range
$\left(\text{\u2212}\infty ,\infty \right),$ and satisfies
${\text{log}}_{b}\left(x\right)=y\phantom{\rule{0.2em}{0ex}}\text{if and only if}\phantom{\rule{0.2em}{0ex}}{b}^{y}=x.$
The most commonly used logarithmic function is the function
${\text{log}}_{e}.$ Since this function uses natural
$e$ as its base, it is called the
natural logarithm . Here we use the notation
$\text{ln}(x)$ or
$\text{ln}\phantom{\rule{0.1em}{0ex}}x$ to mean
${\text{log}}_{e}\left(x\right).$ For example,
and their graphs are symmetric about the line
$y=x$ (
[link] ).
At this
site you can see an example of a base-10 logarithmic scale.
In general, for any base
$b>0,b\ne 1,$ the function
$g\left(x\right)={\text{log}}_{b}(x)$ is symmetric about the line
$y=x$ with the function
$f\left(x\right)={b}^{x}.$ Using this fact and the graphs of the exponential functions, we graph functions
${\text{log}}_{b}$ for several values of
$b>1$ (
[link] ).
Before solving some equations involving exponential and logarithmic functions, let’s review the basic properties of logarithms.
Rule: properties of logarithms
If
$a,b,c>0,b\ne 1,$ and
$r$ is any real number, then
Now we can solve the quadratic equation. Factoring this equation, we obtain
$\left({e}^{x}-3\right)\left({e}^{x}-2\right)=0.$
Therefore, the solutions satisfy
${e}^{x}=3$ and
${e}^{x}=2.$ Taking the natural logarithm of both sides gives us the solutions
$x=\text{ln}\phantom{\rule{0.1em}{0ex}}3,\text{ln}\phantom{\rule{0.1em}{0ex}}2.$
The solution is
$x={10}^{4\text{/}3}=10\sqrt[3]{10}.$
Using the power property of logarithmic functions, we can rewrite the equation as
$\text{ln}\left(2x\right)-\text{ln}\left({x}^{6}\right)=0.$ Using the quotient property, this becomes
$\text{ln}\left(\frac{2}{{x}^{5}}\right)=0.$
Therefore,
$2\text{/}{x}^{5}=1,$ which implies
$x=\sqrt[5]{2}.$ We should then check for any extraneous solutions.
tangent line at a point/range on a function f(x) making f'(x)
Luis
Principles of definite integration?
ROHIT
For tangent they'll usually give an x='s value. In that case, solve for y, keep the ordered pair. then find f(x) prime. plug the given x value into the prime and the solution is the slope of the tangent line. Plug the ordered pair into the derived function in y=mx+b format as x and y to solve for B
Anastasia
parcing an area trough a function f(x)
Efrain
Find the length of the arc y = x^2 over 3 when x = 0 and x = 2.
what's career can one specialize in by doing pure maths
Lucy
Lucy Omollo...... The World is Yours by specializing in pure math. Analytics, Financial engineering ,programming, education, combinatorial mathematics, Game Theory. your skill-set will be like water a necessary element of survival.
mathematics seems to be anthropocentric deductive reasoning and a little high order logic. I only say this because I can only find two things going on which is infinitely smaller than 0 and anything over 1
David
More comprehensive list here: ***onetonline.org/find/descriptor/result/1.A.1.c.1?a=1
Bruce
so how can we differentiate inductive reasoning and deductive reasoning
Henry
thanks very much Mr David
Henry
hi everyone
Sabir
is there anyone who can guide me in learning the mathematics easily
Sabir
Hi Sabir
first step of learning mathematics is by falling in love with it and secondly, watch videos on simple algebra then read and solved problems on it
Leo
yes sabir just do every time practice that is the solution
Henry
it will be work over to you ,u know how mind work ,it prossed the information easily when u are practising regularly