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Solve the following exponential question: a) 2^x = 0.125 b) 3^(2−x) = 243 c) 10^x = 1/10,000 d) 10^x = 1/0.001

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Answers

4 parts
a

x=−3x = -3

b

x=−5x = -5

c

x=−4x = -4

d

x=3x = 3

Step-by-step solution

16 steps

Here are the solutions for the exponential equations.

Solve the following exponential question.

Part a· 4 steps

2x=0.1252^x = 0.125

  1. Convert the decimal to a fraction.

    0.125=1251000=180.125 = \frac{125}{1000} = \frac{1}{8}

  2. Rewrite the equation with the fraction.

    2x=182^x = \frac{1}{8}

  3. Express the right side as a power of 2.

    18=123=2−3\frac{1}{8} = \frac{1}{2^3} = 2^{-3}

  4. Equate the exponents.

    2x=2−32^x = 2^{-3} x=−3x = -3 x=−3\boxed{x = -3}

Part b

x=−5x = -5

3−x=2433^{-x} = 243

  1. Express 243 as a power of 3.

    243=35243 = 3^5

  2. Rewrite the equation.

    3−x=353^{-x} = 3^5

  3. Equate the exponents.

    −x=5-x = 5 x=−5x = -5 x=−5\boxed{x = -5}

Part c

x=−4x = -4

10x=110,00010^x = \frac{1}{10,000}

  1. Express 10,000 as a power of 10.

    10,000=10410,000 = 10^4

  2. Rewrite the equation.

    10x=110410^x = \frac{1}{10^4}

  3. Use the negative exponent rule.

    1104=10−4\frac{1}{10^4} = 10^{-4}

  4. Equate the exponents.

    10x=10−410^x = 10^{-4} x=−4x = -4 x=−4\boxed{x = -4}

Part d

x=3x = 3

10x=10.00110^x = \frac{1}{0.001}

  1. Convert the decimal to a fraction.

    0.001=110000.001 = \frac{1}{1000}

  2. Substitute the fraction into the equation.

    10x=11100010^x = \frac{1}{\frac{1}{1000}}

  3. Simplify the right side.

    10x=100010^x = 1000

  4. Express 1000 as a power of 10.

    1000=1031000 = 10^3

  5. Equate the exponents.

    10x=10310^x = 10^3 x=3x = 3 x=3\boxed{x = 3}
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