10) solve the exponential equation.\n\n81^{2x + 5}=3^{-2x}\n\n11) solve the exponential equation.\n\n4cdot18^…

Explanation:

Step1: Rewrite 81 as (3^4)

$$81^{2x + 5}=(3^4)^{2x+5}=3^{4(2x + 5)}=3^{8x+20}$$ The equation (81^{2x + 5}=3^{-2x}) becomes (3^{8x + 20}=3^{-2x})

Step2: Set exponents equal

Since (a^m=a^n) implies (m = n) for (a>0,a\neq1), we have (8x+20=-2x)

Step3: Solve for (x)

Add (2x) to both sides: (8x+2x+20=-2x + 2x), (10x+20 = 0) Subtract 20: (10x=-20) Divide by 10: (x=\frac{-20}{10}=-2)

Step4: Solve (4\cdot18^{x - 8}+3 = 59)

Subtract 3 from both sides: (4\cdot18^{x-8}=59 - 3=56) Divide by 4: (18^{x - 8}=\frac{56}{4}=14) Take the natural logarithm of both sides: (\ln(18^{x - 8})=\ln(14)) Using the property (\ln(a^b)=b\ln(a)), we get ((x - 8)\ln(18)=\ln(14))

Step5: Solve for (x)

(x-8=\frac{\ln(14)}{\ln(18)}) (x = 8+\frac{\ln(14)}{\ln(18)}\approx8+\frac{2.63906}{2.89037}\approx8 + 0.913\approx8.913)

Answer:

For (81^{2x + 5}=3^{-2x}), (x=-2) For (4\cdot18^{x - 8}+3 = 59), (x = 8+\frac{\ln(14)}{\ln(18)}\approx8.913)