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Exponential Equations To solve exponential equations without logarithms, you need to have equations with comparable exponential expressions on either side of the "equals" sign. Then you can compare the powers and solve. Note that if ar = as, then r = s. If there is a way to rewrite expressions with like bases, the exponents of those bases will then be equal to one another.
When the bases are the same, the powers must also be the same for the equation to be true. You can set the powers equal to each other and solve the resulting equation to find an unknown value.Example 10x = 103 x = 3
101−x = 106 1 − x = 6 Converting a Base When the bases are not the same, you will need to convert one or both bases so that they are the same. This often requires some familiarity with squares, cubes, and higher powers of numbers one through nine. Example 56x+1 = 625 56x+1 = 546x + 1 = 4 Converting Both Bases Sometimes, both bases may need to be converted in order to match. Example 42x+2 = 8 Rewrite the problem using the common base of 2: 4 = 22 (22)(2x+2) = 23Simplify powers raised to a power by multiplying the
exponents. 24x + 4 = 23 Once the bases are equal, the exponents may be set equal to one another and the equation solved. 4x + 4 = 3 Working with Fractions and Negative Exponents Negative exponents indicate that a base belongs in the denominator of a fraction. For instance, = 5-2Example 53x+1 = 53x+1 = 5-23x + 1 = -2 Working with Radical Signs (Square Roots) Note that a square root of a number is the same as the base raised to power of one-half. Example62x−2 = 61/22x − 2 =
(xa)b = xab A power applied to a product within parentheses affects each element inside the parentheses. (xy3)2 =
x2y6
(x + y2)3 does not equal x3 + y6
Example x2 − 3x = 4 How do you solve exponential equations without common bases?In general we can solve exponential equations whose terms do not have like bases in the following way:. Apply the logarithm to both sides of the equation. If one of the terms in the equation has base 10 , use the common logarithm. ... . Use the rules of logarithms to solve for the unknown.. What is the key in solving exponential equation?To solve an exponential equation, take the log of both sides, and solve for the variable. Ln(80) is the exact answer and x=4.38202663467 is an approximate answer because we have rounded the value of Ln(80)..
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