Math / Algebra

QuestionSolve for $x$ in the equation below. Round your answer to the nearest hundredth. Do not round any intermediate computations. $4^{x+9}=15$

Studdy Solution

STEP 1

1. The equation $4^{(x + 9)} = 15$ is exponential.
2. We will use logarithms to solve for $x$ .

STEP 2

1. Take the logarithm of both sides.
2. Use logarithmic properties to solve for $x$ .
3. Calculate the value of $x$ and round to the nearest hundredth.

STEP 3

Take the natural logarithm (or any logarithm, but we'll use natural logarithm here) of both sides of the equation to bring the exponent down:
$\ln(4^{(x + 9)}) = \ln(15)$

STEP 4

Use the power rule of logarithms, which states that $\ln(a^b) = b \cdot \ln(a)$ , to move the exponent in front:
$(x + 9) \cdot \ln(4) = \ln(15)$

STEP 5

Solve for $x + 9$ by dividing both sides by $\ln(4)$ :
$x + 9 = \frac{\ln(15)}{\ln(4)}$

STEP 6

Isolate $x$ by subtracting 9 from both sides:
$x = \frac{\ln(15)}{\ln(4)} - 9$

STEP 7

Calculate the value of $x$ using a calculator. First, compute $\frac{\ln(15)}{\ln(4)}$ , then subtract 9. Round the final answer to the nearest hundredth:
$x \approx \frac{\ln(15)}{\ln(4)} - 9 \approx 0.95315 - 9 \approx -8.05$
The value of $x$ rounded to the nearest hundredth is:
$\boxed{-8.05}$

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QuestionSolve for $x$ in the equation below. Round your answer to the nearest hundredth. Do not round any intermediate computations. $4^{x+9}=15$

Studdy Solution

STEP 1

1. The equation $4^{(x + 9)} = 15$ is exponential.
2. We will use logarithms to solve for $x$ .

STEP 2

1. Take the logarithm of both sides.
2. Use logarithmic properties to solve for $x$ .
3. Calculate the value of $x$ and round to the nearest hundredth.

STEP 3

Take the natural logarithm (or any logarithm, but we'll use natural logarithm here) of both sides of the equation to bring the exponent down:
$\ln(4^{(x + 9)}) = \ln(15)$

STEP 4

Use the power rule of logarithms, which states that $\ln(a^b) = b \cdot \ln(a)$ , to move the exponent in front:
$(x + 9) \cdot \ln(4) = \ln(15)$

STEP 5

Solve for $x + 9$ by dividing both sides by $\ln(4)$ :
$x + 9 = \frac{\ln(15)}{\ln(4)}$

STEP 6

Isolate $x$ by subtracting 9 from both sides:
$x = \frac{\ln(15)}{\ln(4)} - 9$

STEP 7

Calculate the value of $x$ using a calculator. First, compute $\frac{\ln(15)}{\ln(4)}$ , then subtract 9. Round the final answer to the nearest hundredth:
$x \approx \frac{\ln(15)}{\ln(4)} - 9 \approx 0.95315 - 9 \approx -8.05$
The value of $x$ rounded to the nearest hundredth is:
$\boxed{-8.05}$

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QuestionSolve for xxx in the equation below. Round your answer to the nearest hundredth. Do not round any intermediate computations. 4x+9=154^{x+9}=154x+9=15

Studdy Solution

STEP 1

1. The equation 4(x+9)=15 4^{(x + 9)} = 15 4(x+9)=15 is exponential.2. We will use logarithms to solve for x x x.

STEP 2

1. Take the logarithm of both sides.2. Use logarithmic properties to solve for x x x.3. Calculate the value of x x x and round to the nearest hundredth.

STEP 3

Take the natural logarithm (or any logarithm, but we'll use natural logarithm here) of both sides of the equation to bring the exponent down:ln⁡(4(x+9))=ln⁡(15) \ln(4^{(x + 9)}) = \ln(15) ln(4(x+9))=ln(15)

STEP 4

Use the power rule of logarithms, which states that ln⁡(ab)=b⋅ln⁡(a)\ln(a^b) = b \cdot \ln(a)ln(ab)=b⋅ln(a), to move the exponent in front:(x+9)⋅ln⁡(4)=ln⁡(15) (x + 9) \cdot \ln(4) = \ln(15) (x+9)⋅ln(4)=ln(15)

STEP 5

Solve for x+9 x + 9 x+9 by dividing both sides by ln⁡(4)\ln(4)ln(4):x+9=ln⁡(15)ln⁡(4) x + 9 = \frac{\ln(15)}{\ln(4)} x+9=ln(4)ln(15)​

STEP 6

Isolate x x x by subtracting 9 from both sides:x=ln⁡(15)ln⁡(4)−9 x = \frac{\ln(15)}{\ln(4)} - 9 x=ln(4)ln(15)​−9

STEP 7

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QuestionSolve for $x$ in the equation below. Round your answer to the nearest hundredth. Do not round any intermediate computations. $4^{x+9}=15$

1. The equation $4^{(x + 9)} = 15$ is exponential.
2. We will use logarithms to solve for $x$ .

1. Take the logarithm of both sides.
2. Use logarithmic properties to solve for $x$ .
3. Calculate the value of $x$ and round to the nearest hundredth.

Take the natural logarithm (or any logarithm, but we'll use natural logarithm here) of both sides of the equation to bring the exponent down:
$\ln(4^{(x + 9)}) = \ln(15)$

Use the power rule of logarithms, which states that $\ln(a^b) = b \cdot \ln(a)$ , to move the exponent in front:
$(x + 9) \cdot \ln(4) = \ln(15)$

Solve for $x + 9$ by dividing both sides by $\ln(4)$ :
$x + 9 = \frac{\ln(15)}{\ln(4)}$

Isolate $x$ by subtracting 9 from both sides:
$x = \frac{\ln(15)}{\ln(4)} - 9$