For any algebraic expressions
and
and any positive real number
where
if and only if
Definition of a logarithm
For any algebraic expression
S and positive real numbers
and
where
if and only if
One-to-one property for logarithmic functions
For any algebraic expressions
S and
T and any positive real number
where
if and only if
Key concepts
We can solve many exponential equations by using the rules of exponents to rewrite each side as a power with the same base. Then we use the fact that exponential functions are one-to-one to set the exponents equal to one another and solve for the unknown.
When we are given an exponential equation where the bases are explicitly shown as being equal, set the exponents equal to one another and solve for the unknown. See
[link] .
When we are given an exponential equation where the bases are
not explicitly shown as being equal, rewrite each side of the equation as powers of the same base, then set the exponents equal to one another and solve for the unknown. See
[link] ,
[link] , and
[link] .
When an exponential equation cannot be rewritten with a common base, solve by taking the logarithm of each side. See
[link] .
We can solve exponential equations with base
by applying the natural logarithm of both sides because exponential and logarithmic functions are inverses of each other. See
[link] and
[link] .
After solving an exponential equation, check each solution in the original equation to find and eliminate any extraneous solutions. See
[link] .
When given an equation of the form
where
is an algebraic expression, we can use the definition of a logarithm to rewrite the equation as the equivalent exponential equation
and solve for the unknown. See
[link] and
[link] .
We can also use graphing to solve equations with the form
We graph both equations
and
on the same coordinate plane and identify the solution as the
x- value of the intersecting point. See
[link] .
When given an equation of the form
where
and
are algebraic expressions, we can use the one-to-one property of logarithms to solve the equation
for the unknown. See
[link] .
Combining the skills learned in this and previous sections, we can solve equations that model real world situations, whether the unknown is in an exponent or in the argument of a logarithm. See
[link] .
Section exercises
Verbal
How can an exponential equation be solved?
Determine first if the equation can be rewritten so that each side uses the same base. If so, the exponents can be set equal to each other. If the equation cannot be rewritten so that each side uses the same base, then apply the logarithm to each side and use properties of logarithms to solve.
A golfer on a fairway is 70 m away from the green, which sits below the level of the fairway by 20 m. If the golfer hits the ball at an angle of 40° with an initial speed of 20 m/s, how close to the green does she come?
A mouse of mass 200 g falls 100 m down a vertical mine shaft and lands at the bottom with a speed of 8.0 m/s. During its fall, how much work is done on the mouse by air resistance
Chemistry is a branch of science that deals with the study of matter,it composition,it structure and the changes it undergoes
Adjei
please, I'm a physics student and I need help in physics
Adjanou
chemistry could also be understood like the sexual attraction/repulsion of the male and female elements. the reaction varies depending on the energy differences of each given gender. + masculine -female.
Pedro
A ball is thrown straight up.it passes a 2.0m high window 7.50 m off the ground on it path up and takes 1.30 s to go past the window.what was the ball initial velocity
2. A sled plus passenger with total mass 50 kg is pulled 20 m across the snow (0.20) at constant velocity by a force directed 25° above the horizontal. Calculate (a) the work of the applied force, (b) the work of friction, and (c) the total work.
you have been hired as an espert witness in a court case involving an automobile accident. the accident involved car A of mass 1500kg which crashed into stationary car B of mass 1100kg. the driver of car A applied his brakes 15 m before he skidded and crashed into car B. after the collision, car A s
can someone explain to me, an ignorant high school student, why the trend of the graph doesn't follow the fact that the higher frequency a sound wave is, the more power it is, hence, making me think the phons output would follow this general trend?
Nevermind i just realied that the graph is the phons output for a person with normal hearing and not just the phons output of the sound waves power, I should read the entire thing next time
Joseph
Follow up question, does anyone know where I can find a graph that accuretly depicts the actual relative "power" output of sound over its frequency instead of just humans hearing
Joseph
"Generation of electrical energy from sound energy | IEEE Conference Publication | IEEE Xplore" ***ieeexplore.ieee.org/document/7150687?reload=true
A string is 3.00 m long with a mass of 5.00 g. The string is held taut with a tension of 500.00 N applied to the string. A pulse is sent down the string. How long does it take the pulse to travel the 3.00 m of the string?
