Solving Exponential and Logarithmic Equations

Solving Exponential and Logarithm Equations:

Inverses:

Keep in mind that the exponential and logarithmic functions are one-to-one functions. That signifies that they have inverses. As well remember that when inverses are composed with one other, they inverse out and just the argument is returned. We are going to utilize that to our advantage to help in solving logarithmic and exponential equations.

Please remember the given facts:

i) loga ax = x
ii) log 10x = x
iii) ln ex = x
iv) a loga x = x
v) 10log x = x
vi) eln x = x

Solving Exponential Equations Algebraically:

a) Isolate exponential expression on one side.

b) Take the logarithm of both sides. The base for logarithm must be similar as the base in an exponential expression. Otherwise, if you are only concerned in a decimal approximation, you might take the natural log or common log of both the sides (in effect employing the change of base formula).

c) Solve for the variable.

d) Ensure your answer. It might be possible to get answers that do not check. Generally, the answer will include complex numbers whenever this happens, since the domain of the exponential function is all reals.

Solving Logarithmic Equations Algebraically:

a) Employ the properties of logarithms to join the sum, difference, and/or constant multiples of logarithms to a single logarithm.

b) Apply an exponential function to both the sides. The base employed in the exponential function must be similar as the base in logarithmic function. The other way of executing this task is to write the logarithmic equation in an exponential form.

c) Solve for variable.

d) Check your answer. It might be possible to introduce the extraneous solutions. Make sure that whenever you plug your answer back to the arguments of the logarithms in original equation, then the arguments are all positive. Keep in mind; that you can only take the log of positive number.

Solving Equations Graphically:

Sometimes, it is not possible to solve an equation comprising logarithms or exponential functions algebraically. This is in particular true whenever the equation comprises transcendental (logs and/or exponentials) and algebraic components. In cases similar to these, it might be essential to employ the graphing calculator to help in finding the solution to the equation.

a) Rewrite the equation and hence all the terms are on one side.
b) Graph the expression.
c) Utilize the Root or Zero function beneath the Calc menu.

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