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Chapter 8: Problem 14

Solve (a) \(10^{\log x}=17\) (b) \(10^{2 \log x}=17\) (c) \(10^{x} 10^{2 x}=90\) (d) \(10^{2 x}=30\left(10^{2}\right)\)

Short Answer

Expert verified
a) \(10^{\log x}=17\) b) \(10^{2 \log x}=17\) c) \(10^{x} 10^{2 x}=90\) d) \(10^{2 x}=30\left(10^{2}\right)\) Answer: The solutions for the given equations are as follows: a) \(x = 17\) b) \(x = \sqrt{17}, -\sqrt{17}\) c) \(x = \frac{\log(90)}{3}\) d) \(x = 1\)

Step by step solution

01

Use logarithm property

Recall that \(a^{\log_a x} = x\). This property can be applied here, since our base is 10. We have \(10^{\log x} = x\).
02

Substitute and solve for x

Substitute this property into the given equation: \(x = 17\). Therefore, the solution to this equation is \(x=17\). ##Equation (b) \(10^{2 \log x}=17\)##
03

Use logarithm property

Recall the property \(a^{n \log_a x } = x^n\). Apply this property here, with \(n=2\). We have \(10^{2\log x} = x^2\).
04

Substitute and solve for x

Substitute this property into the given equation: \(x^2 = 17\). Taking the square root of both sides, we obtain \(x = \pm\sqrt{17}\). Therefore, the solutions to this equation are \(x=\sqrt{17}, -\sqrt{17}\). ##Equation (c) \(10^{x} 10^{2 x}=90\)##
05

Use exponent property

Recall the property \(a^m \cdot a^n = a^{m+n}\). Apply this property here to combine the exponential terms: \(10^{x+2x} = 10^{3x}\).
06

Take logarithm of both sides

Take the logarithm base 10 of both sides: \(\log(10^{3x}) = \log(90)\).
07

Use logarithm property and solve for x

Recall the property \(\log_a (a^x) = x\). Apply this property to the left side of the equation: \(3x = \log(90)\). Divide both sides by 3: \(x = \frac{\log(90)}{3}\). Therefore, the solution to this equation is \(x = \frac{\log(90)}{3}\). ##Equation (d) \(10^{2 x}=30\left(10^{2}\right)\)##
08

Simplify the equation

Remove parentheses: \(10^{2x} = 30 \cdot 10^2\).
09

Divide both sides by 30

Divide both sides by 30 to isolate the exponential term: \(10^{2x} = 10^2\).
10

Take logarithm of both sides

Take the logarithm base 10 of both sides: \(\log(10^{2x}) = \log(10^2)\).
11

Use logarithm property and solve for x

Recall the property \(\log_a (a^x) = x\). Apply this property to both sides of the equation: \(2x = 2\). Divide both sides by 2: \(x = 1\). Therefore, the solution to this equation is \(x = 1\).

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