Using the definition of a logarithm to solve logarithmic equations
For any algebraic expression
$\text{\hspace{0.17em}}S\text{\hspace{0.17em}}$ and real numbers
$\text{\hspace{0.17em}}b\text{\hspace{0.17em}}$ and
$\text{\hspace{0.17em}}c,$ where
$\text{\hspace{0.17em}}b>0,\text{}b\ne 1,$
[link] represents the graph of the equation. On the graph, the
x -coordinate of the point at which the two graphs intersect is close to 20. In other words
$\text{\hspace{0.17em}}{e}^{3}\approx 20.\text{\hspace{0.17em}}$ A calculator gives a better approximation:
$\text{\hspace{0.17em}}{e}^{3}\approx \mathrm{20.0855.}$
Use a graphing calculator to estimate the approximate solution to the logarithmic equation
$\text{\hspace{0.17em}}{2}^{x}=1000\text{\hspace{0.17em}}$ to 2 decimal places.
Using the one-to-one property of logarithms to solve logarithmic equations
As with exponential equations, we can use the one-to-one property to solve logarithmic equations. The one-to-one property of logarithmic functions tells us that, for any real numbers
$\text{\hspace{0.17em}}x>0,$$S>0,$$T>0\text{\hspace{0.17em}}$ and any positive real number
$\text{\hspace{0.17em}}b,$ where
$\text{\hspace{0.17em}}b\ne 1,$
So, if
$\text{\hspace{0.17em}}x-1=8,$ then we can solve for
$\text{\hspace{0.17em}}x,$ and we get
$\text{\hspace{0.17em}}x=9.\text{\hspace{0.17em}}$ To check, we can substitute
$\text{\hspace{0.17em}}x=9\text{\hspace{0.17em}}$ into the original equation:
$\text{\hspace{0.17em}}{\mathrm{log}}_{2}\left(9-1\right)={\mathrm{log}}_{2}\left(8\right)=3.\text{\hspace{0.17em}}$ In other words, when a logarithmic equation has the same base on each side, the arguments must be equal. This also applies when the arguments are algebraic expressions. Therefore, when given an equation with logs of the same base on each side, we can use rules of logarithms to rewrite each side as a single logarithm. Then we use the fact that logarithmic functions are one-to-one to set the arguments equal to one another and solve for the unknown.
For example, consider the equation
$\text{\hspace{0.17em}}\mathrm{log}\left(3x-2\right)-\mathrm{log}\left(2\right)=\mathrm{log}\left(x+4\right).\text{\hspace{0.17em}}$ To solve this equation, we can use the rules of logarithms to rewrite the left side as a single logarithm, and then apply the one-to-one property to solve for
$\text{\hspace{0.17em}}x:$
hi can you give another equation I'd like to solve it
Daniel
what is the value of x in 4x-2+3
Olaiya
if 4x-2+3 = 0
then
4x = 2-3
4x = -1
x = -(1÷4) is the answer.
Jacob
4x-2+3
4x=-3+2
4×=-1
4×/4=-1/4
LUTHO
then x=-1/4
LUTHO
4x-2+3
4x=-3+2
4x=-1
4x÷4=-1÷4
x=-1÷4
LUTHO
A research student is working with a culture of bacteria that doubles in size every twenty minutes. The initial population count was 1350 bacteria. Rounding to five significant digits, write an exponential equation representing this situation. To the nearest whole number, what is the population size after 3 hours?
A guy wire for a suspension bridge runs from the ground diagonally to the top of the closest pylon to make a triangle. We can use the Pythagorean Theorem to find the length of guy wire needed. The square of the distance between the wire on the ground and the pylon on the ground is 90,000 feet. The square of the height of the pylon is 160,000 feet. So, the length of the guy wire can be found by evaluating √(90000+160000). What is the length of the guy wire?