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Solve the exponential equation. Write the exact solution and the approximation to four decimal places. Type "DNE" if there is no solution.

[tex]\[ 2 \cdot 10^{x+4} = -1 \][/tex]

1. Exact solution: [tex]$\square$[/tex]
2. Approximation: [tex]$\square$[/tex]

Sagot :

GinnyAnsweridn
Let's solve the given exponential equation step by step:

The equation given is:

[tex]\[ 2 \cdot 10^{x+4} = -1 \][/tex]

Step 1: Analyze the equation. Notice that [tex]\(10^{x+4}\)[/tex] refers to an exponential function with the base 10. This function is always positive because any power of 10 (or any positive number) is positive.

Step 2: Multiply this positive quantity [tex]\(10^{x+4}\)[/tex] by 2, which still keeps the result positive, since multiplying a positive number by a positive number results in a positive number.

Step 3: Observe the right-hand side of the equation, which is [tex]\(-1\)[/tex]. This is a negative number.

Given that the left side of the equation (which is a positive number) is set equal to a negative number ([tex]\(-1\)[/tex]), we see a fundamental conflict: a positive number can never be equal to a negative number.

Thus, there is no value of [tex]\(x\)[/tex] that can satisfy this equation.

Therefore, the exact solution is:

Exact solution: DNE

Since the equation does not have any solution, the approximation also does not exist.

Approximation: DNE
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