Solve 1/7*3^x 1 1/2*3^x^2 = 23/14*3^5

Question

Solve the equation

x = \frac{4 + lo g _{3} ( 23 )}{2}

Alternative Form

x \approx 3.427025

Evaluate

\frac{1}{7} \times 3^{x} \times 1 \frac{1}{2} \times 3^{x} = \frac{23}{14} \times 3^{5}

Simplify

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Evaluate

\frac{1}{7} \times 3^{x} \times 1 \frac{1}{2} \times 3^{x}

Covert the mixed number to an improper fraction

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Convert the expressions

1 \frac{1}{2}

Multiply the denominator of the fraction by the whole number and add the numerator of the fraction

\frac{2 + 1}{2}

Add the terms

\frac{3}{2}

\frac{1}{7} \times 3^{x} \times \frac{3}{2} \times 3^{x}

Multiply the terms

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Evaluate

\frac{1}{7} \times \frac{3}{2}

To multiply the fractions,multiply the numerators and denominators separately

\frac{3}{7 \times 2}

Multiply the numbers

\frac{3}{14}

\frac{3}{14} \times 3^{x} \times 3^{x}

Multiply the terms with the same base by adding their exponents

\frac{3}{14} \times 3^{x + x}

Add the terms

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Evaluate

x + x

Collect like terms by calculating the sum or difference of their coefficients

(1 + 1) x

Add the numbers

2 x

\frac{3}{14} \times 3^{2 x}

Multiply the terms

\frac{3 \times 3 ^{2 x}}{14}

Multiply the terms

\frac{3 ^{2 x + 1}}{14}

\frac{3 ^{2 x + 1}}{14} = \frac{23}{14} \times 3^{5}

Multiply the numbers

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Evaluate

\frac{23}{14} \times 3^{5}

Multiply the numbers

\frac{23 \times 3 ^{5}}{14}

Multiply the numbers

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Evaluate

23 \times 3^{5}

Evaluate the power

23 \times 243

Multiply the numbers

5589

\frac{5589}{14}

\frac{3 ^{2 x + 1}}{14} = \frac{5589}{14}

Multiply both sides of the equation by 14

\frac{3 ^{2 x + 1}}{14} \times 14 = \frac{5589}{14} \times 14

Multiply the terms

3^{2 x + 1} = 5589

Take the logarithm of both sides

lo g_{3} (3^{2 x + 1}) = lo g_{3} (5589)

Evaluate the logarithm

2 x + 1 = lo g_{3} (5589)

Move the constant to the right-hand side and change its sign

2 x = lo g_{3} (5589) - 1

Divide both sides

\frac{2 x}{2} = \frac{lo g _{3} ( 5589 ) - 1}{2}

Divide the numbers

x = \frac{lo g _{3} ( 5589 ) - 1}{2}

Solution

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Evaluate

lo g_{3} (5589) - 1

Simplify

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Evaluate

lo g_{3} (5589)

Use lo g_{a} (x \times y) = lo g_{a} (x) + lo g_{a} (y) to transform the expression

lo g_{3} (243) + lo g_{3} (23)

Simplify the expression

5 + lo g_{3} (23)

5 + lo g_{3} (23) - 1

Calculate

4 + lo g_{3} (23)

x = \frac{4 + lo g _{3} ( 23 )}{2}

Alternative Form

x \approx 3.427025

Show Solution

Solve the equation

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