Pregunta
Use the like bases property to solve the equation \( 10^{x+4}=10^{8 x+9} \) \( x= \)
Ask by Beck Best. in the United States
Mar 13,2025
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\( x = -\frac{5}{7} \)
Solución
We are given the equation:
\[
10^{x+4} = 10^{8x+9}
\]
Since the bases are the same, we can set the exponents equal to each other:
\[
x + 4 = 8x + 9
\]
Subtract \( x \) from both sides:
\[
4 = 7x + 9
\]
Subtract \( 9 \) from both sides:
\[
4 - 9 = 7x \quad \Rightarrow \quad -5 = 7x
\]
Solve for \( x \) by dividing both sides by \( 7 \):
\[
x = -\frac{5}{7}
\]
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To solve the equation \( 10^{x+4} = 10^{8x+9} \), we can use the fact that if the bases are equal, then the exponents must also be equal. This gives us the equation: \[ x + 4 = 8x + 9. \] Now, let's rearrange the equation to isolate \( x \). Subtract \( x \) from both sides: \[ 4 = 7x + 9. \] Next, subtract 9 from both sides: \[ 4 - 9 = 7x, \] which simplifies to: \[ -5 = 7x. \] Finally, divide both sides by 7: \[ x = -\frac{5}{7}. \] So the solution is: \[ x = -\frac{5}{7}. \]
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