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To solve the equation [tex]\( 25^x - 6 \times 5^{x+5} = 0 \)[/tex], let's break it down step-by-step.
### Step 1: Simplify the Terms
First, observe that [tex]\( 25 \)[/tex] can be expressed as a power of [tex]\( 5 \)[/tex]:
[tex]\[ 25 = 5^2 \][/tex]
So we can rewrite [tex]\( 25^x \)[/tex] as:
[tex]\[ 25^x = (5^2)^x = 5^{2x} \][/tex]
Substituting this into the original equation:
[tex]\[ 5^{2x} - 6 \times 5^{x+5} = 0 \][/tex]
### Step 2: Factor Common Terms
Next, recall the properties of exponents. We can split [tex]\( 5^{x+5} \)[/tex] as:
[tex]\[ 5^{x+5} = 5^x \times 5^5 \][/tex]
So the equation becomes:
[tex]\[ 5^{2x} - 6 \times 5^x \times 5^5 = 0 \][/tex]
Since [tex]\( 5^5 = 3125 \)[/tex], the equation now is:
[tex]\[ 5^{2x} - 6 \times 3125 \times 5^x = 0 \][/tex]
### Step 3: Substitute to Solve the Equation
Let [tex]\( y = 5^x \)[/tex]. This transformation makes the equation easier to handle:
[tex]\[ y^{2} - 6 \times 3125 \times y = 0 \][/tex]
### Step 4: Further Simplify and Factor
Combine the constant term:
[tex]\[ y^2 - 18750y = 0 \][/tex]
Factor out [tex]\( y \)[/tex]:
[tex]\[ y(y - 18750) = 0 \][/tex]
### Step 5: Solve the Factored Equation
Set each factor equal to zero:
[tex]\[ y = 0 \quad \text{or} \quad y - 18750 = 0 \][/tex]
So, we have:
[tex]\[ y = 0 \quad \text{or} \quad y = 18750 \][/tex]
### Step 6: Revert the Substitution
Recall that [tex]\( y = 5^x \)[/tex]. Therefore, we should solve:
[tex]\[ 5^x = 0 \quad \text{or} \quad 5^x = 18750 \][/tex]
Since [tex]\( 5^x = 0 \)[/tex] has no real solution (as any power of 5, a positive number, is always positive), we focus on:
[tex]\[ 5^x = 18750 \][/tex]
### Step 7: Solve for [tex]\( x \)[/tex]
To solve for [tex]\( x \)[/tex], we take the logarithm base 5 of both sides:
[tex]\[ x = \log_{5}(18750) \][/tex]
### Final Solution
So, the solution to the equation [tex]\( 25^x - 6 \times 5^{x+5} = 0 \)[/tex] is:
[tex]\[ x = \frac{\log(18750)}{\log(5)} \][/tex]
This expresses [tex]\( x \)[/tex] in terms of common logarithms.
### Step 1: Simplify the Terms
First, observe that [tex]\( 25 \)[/tex] can be expressed as a power of [tex]\( 5 \)[/tex]:
[tex]\[ 25 = 5^2 \][/tex]
So we can rewrite [tex]\( 25^x \)[/tex] as:
[tex]\[ 25^x = (5^2)^x = 5^{2x} \][/tex]
Substituting this into the original equation:
[tex]\[ 5^{2x} - 6 \times 5^{x+5} = 0 \][/tex]
### Step 2: Factor Common Terms
Next, recall the properties of exponents. We can split [tex]\( 5^{x+5} \)[/tex] as:
[tex]\[ 5^{x+5} = 5^x \times 5^5 \][/tex]
So the equation becomes:
[tex]\[ 5^{2x} - 6 \times 5^x \times 5^5 = 0 \][/tex]
Since [tex]\( 5^5 = 3125 \)[/tex], the equation now is:
[tex]\[ 5^{2x} - 6 \times 3125 \times 5^x = 0 \][/tex]
### Step 3: Substitute to Solve the Equation
Let [tex]\( y = 5^x \)[/tex]. This transformation makes the equation easier to handle:
[tex]\[ y^{2} - 6 \times 3125 \times y = 0 \][/tex]
### Step 4: Further Simplify and Factor
Combine the constant term:
[tex]\[ y^2 - 18750y = 0 \][/tex]
Factor out [tex]\( y \)[/tex]:
[tex]\[ y(y - 18750) = 0 \][/tex]
### Step 5: Solve the Factored Equation
Set each factor equal to zero:
[tex]\[ y = 0 \quad \text{or} \quad y - 18750 = 0 \][/tex]
So, we have:
[tex]\[ y = 0 \quad \text{or} \quad y = 18750 \][/tex]
### Step 6: Revert the Substitution
Recall that [tex]\( y = 5^x \)[/tex]. Therefore, we should solve:
[tex]\[ 5^x = 0 \quad \text{or} \quad 5^x = 18750 \][/tex]
Since [tex]\( 5^x = 0 \)[/tex] has no real solution (as any power of 5, a positive number, is always positive), we focus on:
[tex]\[ 5^x = 18750 \][/tex]
### Step 7: Solve for [tex]\( x \)[/tex]
To solve for [tex]\( x \)[/tex], we take the logarithm base 5 of both sides:
[tex]\[ x = \log_{5}(18750) \][/tex]
### Final Solution
So, the solution to the equation [tex]\( 25^x - 6 \times 5^{x+5} = 0 \)[/tex] is:
[tex]\[ x = \frac{\log(18750)}{\log(5)} \][/tex]
This expresses [tex]\( x \)[/tex] in terms of common logarithms.
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