To solve the equation \( 10 \times 3^x + 3 = 60 \), we can start by isolating the term involving \( x \).
- Subtract 3 from both sides:
\[ 10 \times 3^x = 60 - 3 \] \[ 10 \times 3^x = 57 \]
- Divide both sides by 10:
\[ 3^x = \frac{57}{10} \] \[ 3^x = 5.7 \]
- Take the logarithm of both sides. We can use the natural logarithm or log base 10, but here I'll use natural logarithm:
\[ \ln(3^x) = \ln(5.7) \]
- Apply the logarithmic identity \( \ln(a^b) = b \ln(a) \):
\[ x \ln(3) = \ln(5.7) \]
- Solve for \( x \):
\[ x = \frac{\ln(5.7)}{\ln(3)} \]
Now, we will calculate the values of \( \ln(5.7) \) and \( \ln(3) \):
Using a calculator:
\[ \ln(5.7) \approx 1.7409 \] \[ \ln(3) \approx 1.0986 \]
Substituting these values into the expression for \( x \):
\[ x \approx \frac{1.7409}{1.0986} \approx 1.586 \]
Finally, rounding to three decimal places, we find:
\[ x \approx 1.586 \]
From your response options, it seems I made a mistake initially; I was meant to calculate for negative values. However, given the positive solution derived, it leads us to cross-check the interpretation of the question.
On double-checking the initial calculations: \[ x = \ln\left(\frac{57}{10}\right) / \ln(3) \] leads me to \[ x \approx 1.586 \] But since our given alternate forms specify negatives, it's essential to evaluate which of those might have been misrepresented here.
Please verify if I should be addressing \( x = -1.369, -2.387, -1.796, -2.169 \) which result if the base and structure of original formulation were mispositioned.
For continuous verification, the correct approach is \(-2 \text{ options, or a derivation mismatch.}\)
Would you prefer I recast values in line with opposite terms, checking for invalid placements? Please acknowledge any adjustments and the preferred grid for clarification on negative findings.
3 answers