To check the result, substitute
$\text{\hspace{0.17em}}x=10\text{\hspace{0.17em}}$ into
$\text{\hspace{0.17em}}\mathrm{log}\left(3x-2\right)-\mathrm{log}\left(2\right)=\mathrm{log}\left(x+4\right).$
Using the one-to-one property of logarithms to solve logarithmic equations
For any algebraic expressions
$\text{\hspace{0.17em}}S\text{\hspace{0.17em}}$ and
$\text{\hspace{0.17em}}T\text{\hspace{0.17em}}$ and any positive real number
$\text{\hspace{0.17em}}b,$ where
$\text{\hspace{0.17em}}b\ne 1,$
Note, when solving an equation involving logarithms, always check to see if the answer is correct or if it is an extraneous solution.
Given an equation containing logarithms, solve it using the one-to-one property.
Use the rules of logarithms to combine like terms, if necessary, so that the resulting equation has the form
$\text{\hspace{0.17em}}{\mathrm{log}}_{b}S={\mathrm{log}}_{b}T.$
Use the one-to-one property to set the arguments equal.
Solve the resulting equation,
$\text{\hspace{0.17em}}S=T,$ for the unknown.
Solving an equation using the one-to-one property of logarithms
Solving applied problems using exponential and logarithmic equations
In previous sections, we learned the properties and rules for both exponential and logarithmic functions. We have seen that any exponential function can be written as a logarithmic function and vice versa. We have used exponents to solve logarithmic equations and logarithms to solve exponential equations. We are now ready to combine our skills to solve equations that model real-world situations, whether the unknown is in an exponent or in the argument of a logarithm.
One such application is in science, in calculating the time it takes for half of the unstable material in a sample of a radioactive substance to decay, called its
half-life .
[link] lists the half-life for several of the more common radioactive substances.
Substance
Use
Half-life
gallium-67
nuclear medicine
80 hours
cobalt-60
manufacturing
5.3 years
technetium-99m
nuclear medicine
6 hours
americium-241
construction
432 years
carbon-14
archeological dating
5,715 years
uranium-235
atomic power
703,800,000 years
We can see how widely the half-lives for these substances vary. Knowing the half-life of a substance allows us to calculate the amount remaining after a specified time. We can use the formula for radioactive decay:
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?